Arithmetical Functions (London Mathematical Society Lecture Note Series)

multiplicative.

By definition 1 * µ = E, and so

(a)

1, if n = 1, din µ(d) (b)

The functions

(c)

µ(p)

(d)

t(pk) = k+1,

(e)

(p(pk) =

p(n)

µ,

cp,

t,

tk are multiplicative.

for every prime p, µ(pk) = 0, if k 2 2.2)

1

tm(Pk)

km

Pk

=n

otherwise.

0

jl

-

IT pIn

(

(1 -

in

1-

1

1

).

P-1

P-1).

(f) p(n) is the number of residue-classes mod n, which are coprime with n. this (D(n/d) = u(i'd), where

Proof:

Denoting

number

by

O(n)

for

the

moment3l,

ed = (v 5 n, gcd(v,n) = d) _ (v s n/d, gcd( v, n/d) = 1 The union

1°

dn I

d

).

consists of all Integers in [1,...,n], and is disjoint,

and so n = Edln tt(e d) hence 0=µ*idIN =cp.

Edln D(n/d). Therefore idIN

=

1

*

0;

The next theorem, known as the Mt BIUs Inversion Formula, does not express anything other than the fact that the functions 1IN and µ are Inverse to each other with respect to convolution.

Theorem 2.3. Given two arithmetical functions f and g, the relations

2) 3)

by Induction from (a) and I + t(p) + µ(p2) + ... + µ(pk) = 0. For a [finite] set D we denote by a(te) the number of elements In 2).

Tools from Number Theory

10

f(d) for every n e IN,

g(n) =

(2.10')

d n

and (2.10")

f(n) =

do

µ(d)

g(a) for every n

f IN

are equivalent.

Alternatively, these relations may be expressed in the following way.

Theorem 2.3'. The map T: C Tf :

-) CN, defined by f(d)'

n

Is linear and bijective, with the Inverse map T 1, T -If

This follows from Tf =

1

:

n 3 diZnµ(d) ' f (n ).

* f and T_if = µ * f by the associativity of

convolution.

11

The results just given may be generalized: let h: IN - C be a completely multiplicative function h: IN -* C [this means, that h(1) = 1 and h(n), thus h is a non-trivial homomorphism from h(m ' n) = h(m) the semi-group (N,-) into the semi-group (C,.)]. Then the map

Th: C - CN , (2.12' )

defined by Th f :

nH din

Is linear and bijective, with an Inverse map Th 1 given by (2.12")

Th 1g : n H

h 1(*)

d

=

µ(d).

this is the pointwise product!),

Thf=h*f, andThlg=(rh)*g. A second kind of Monius inversion formula is now presented as follows.

Theorem 2.4. Let h: IN -> C be completely multiplicative and consider the vector-space

1.2.

Arithmetical Functions, Convolution, Mobius Inversion Formula

it

F: [1, w I - C }.

Cdefined by

Then the map 7h : Cti'°°t

.T F: x Hnsx E h(n) h

(2.13')

F(x/n), ( x 2 1

.

),

is linear and bijective, its inverse being

7,-'F: x H nsx Z

(2.13")

µ(n)-h(n)

F(x/n), ( x z I

.

).

proof. Obviously, 7h is linear. If rh F = 0, then for any x z 1,

0 =nsx Z µ(n)'h(n) J' F(x/n) = nsx E

i.i(n)'h(n)

1i

'

x/nk),

ks(x/n)

and, putting t = k'n, this double sum E tsx

v(n) = F(x)

t

h(1),

and so F = 0. Hence 7, is injective. The surJectivity of 7'h is proved similarly. 11

An application of this result is given next. 4) Corollary 2.5. If x z 1, then nsx n-1'µ(n)

(2.14)

proof. Choose h = 1=

nsx

1,

F=

I

I

s 1+ x

1.

in Theorem 2.4. Then 9"hF(x) = [x] and

µ(n)-[x/n] = x

nsx

n-µ(n) + Z nsx

where "%l s 1. The obvious observation result.

I

µ(n)'>4n,

Ensx µ(n) 45n

I

S x gives the

To obtain an impression of the erratic behaviour of the µ-function, see Figure 1.1, which gives values of the MoBius [L-function in the range

1sns298. 4)

We remark that Inequality (2.14), obtained elementarily, Is rather weak. From the prime number theorem (see 1.6) It follows that n-1.µ(n) Is convergent to zero (see also Figure 2, next page).

Tools from Number Theory

12

1

0

u

I

I I SO

III

I

I

II

I

100

200

ISO

25O

Figure 1.1, Values of the Moblus Function

The values of

En,,,,

n-1 µ(n) are plotted in

Figure 2 In the range I s x s 598. The values for x = 1 and x = 2 are cut.

100

200

400

300

Figure 1.2, Values of

Soo

nsN

Finally, we give some results on the divisor function t. For any E > 0 there is a constant C(E), for which the estimate t(n) s C(E)

n'

holds; this is proved, in HARDY-WRIGHT [19S6], Theorem

315,

for

example. A more general result is given in the same work, § 18.1 (see also SCHWARZ 11987a]), as follows

Theorem 2.6. If f is multiplicative and satisfies lim

n-m

lim

f(pk) = 0, then

f(n) = 0.

Proof. Given s>O, there is a constant N(E) such that If(pk)1 < E, if pk z N(s). In particular, lf(pk)l < if pk 2 N(1). Therefore there is 1

1.2. Arithmetical Functions, Convolution, Mobius Inversion Formula

13

some constant y, independent of s, p, k, for which If(pk)I 5 Y. The number of integers, composed entirely from prime-powers pt s N(E), is finite, and so any of these numbers is less than some NP(s). > N(s), which divides n. if n > N*(s), then there is some prime-power Denote by NPP(s) the number of prime-powers below N(E). The function f being multiplicative, we obtain ple

I f(n) I

s Y1vPP{1)

.

I

s,

if n > N*(s).

To obtain an impression of the behaviour of the divisor-function, this function is plotted in the range 1s n s 298 (see Figure 1.3, with the mean - value 20

plotted inversely), and in the range

15

10001 s n s

to

SI

10598

Illl

"ii" i`" _ iii ""ii i ilf " it II I If f I I 1111111 III II II r11'IiI I II I IIII I I I1 Ii 0 I 11'1'1l'19

I

Figure 1.3. The Divisor Function in the range I s n s 298.

So -i

10.100

10.300

(see

Figure 1.4).

I

10.500

Figure 1.4. The Divisor Function In the range 10001 s n s 10598

Tools from Number Theory

14

Theorem 2.7. The following asymptotic formulae are true for the divisor-function t: (a)

Z ti(n) nsx

(b)

nsx

n-1

= x log x + ( 2e -1) x + 0(,r-x), '

t(n) = z logZx +

log x + K + O(1/

2L'

),

with some constant K, (c)

nsx

ti(n) s C($)

x

)2e-1

(1 + log x

for 8 =

1,

2, ... .

Proof. (a) The simple attempt of interchanging the order of summation,

Ensx t(n) = Znsx EdIn 1 = Zdsx lnsx, n=O mod d

= Edsx [x/d] = x

d--1

Zdsx

1

+ O(x),

gives a result that is definitely weaker than (a). But a useful trick, due to DIRICHLET, proves formula (a):

_ dsx/m E 1+E Z t(n) _ E Z 1 = YmsB dsx/B msx/d

nsx

1- > msB

E

dsx/B

choosing the parameter B = x1 (this optimal!), the last line changes into = 2 :msIrX-

1;

is

[X/m] - [ x ]2.

Writing Ex/ml = x/m + O(1) and using the formula for ins (1/n), obtained from EuLER's summation formula (see (1.4), §1), one arrives at (a). (b) follows from (a) by partial summation. :i:

iY:;tiisY:is:iiii:X:F:kci x

Figure I.S. Lattice points below

x

1 .3.

periodic Functions, Even Functions, Ramanujan Sums

15

(c) is proved by induction; the assertion is true for t = 1. Assume that (c) Is true for t. Then, by partial summation, for x 2 1, s C(E)

E nsx

(2.15)

(

1

+ log x )2z.

Using multiplicativity and t(pk) = k+1, we see that for every pair (m,n) of positive integers the Inequality

t Is "sub-multlplicatlve"]. Therefore, writing n =

we

obtain t2+1(n) n&x

_

nsx ti(n) E 2: dsxt"(d)

In

1=

msx/d

dsx

Zmsx/d tE(m).

Using the induction hypothesis for the sum over m and then (2.15), a short calculation gives the assertion.

1.3. PERIODIC FUNCTIONS, EVEN FUNCTIONS, RAMANUJAN SUMS

Definition. Let r be a positive Integer and p a prime. An arithmetical function f is called r-periodic, if f(n+r) = f(n) for every positive Integer n, r-even, if f(n) = f(gcd(n,r)) for every positive integer n, p-fibre-constant, if f(n) = f(p',), where the exponents v are taken from the prime factor decomposition P

n=

qva.

qv' IIn

f is termed periodic [resp. even], if there is some r for which f is r-periodic [resp. r-even]. Obviously, an r-even function Is r-periodic.

Standard examples of

Tools from Number Theory

16

r-periodic functions are the exponential functions e(a), where a = aE7Z,rEIN, and where

a

r'

e(a): n -

(3.1)

These exponential functions satisfy the following orthogonality relations:

Let dir, tir, and gcd(a,d) = gcd(b,t) = r

r

J 0, if d

e` d m) e b m)

1,

m=1

if

Then

1.

t or a+b t and a+b

d

0 mod d 0 mod d.

a

The sum on the left-hand side is r-1

e(m

21

lsmsr

0,

at + bd dt

if

1

(at+bd),

otherwise.

The RAMANU.JAN sum c r is a special exponential sum:

cr(n) _

(3.2)

lsasr, gcd(a,r)=1 exp (2TCi '

r

'n ).

Important properties of the RAMANLUAN sums are given in the following theorem. Theorem 3.1. RAMANL(IAN sums have the following properties.

(a) The RAMANUJAN sum cr is r-periodic. (b)

Cr(n) =

2: dlgcd(r,n)

T

d

(c) The RAMANUJAN sum cr is r-even.

(d) For any fixed n the map r H Cr(n) is multiplicative. (e) The RAMANUJAN sums satisfy the following orthogonality relations:

If tir and dir, then r

(3.3)

I cd(m)

m=1

c (m)

- J

l

t

0, if

d

t,

cp(d), if d

t.

Proof. (a) is obvious. (b) Using * it = E, the value cr(n) is 1

cr(n) = 1sl

( r e\ r

-

n/

ii(d)

1.3. periodic Functions, Even Functions, Ramanujan Sums

Idir t(d) the latter

isasr,a-0 mod d e (r

part of the equation above is equal to

17

n);

.

isbsr/d

e( r/d b

n

and this expression is 0 if (r/d) 4' n, and is equal to r/d otherwise. Therefore,

cr(n) = Zdlr, (r/d)In µ(d)

.

(r/d) = Y-tlr, tln µ(r/t)

.

t.

(c) The r-evenness of the map n H cr(n) is obvious from (b). (d) In (b) cr(n) = (µ * Fn )(r) was obtained, where t, if tin,

0 otherwise.

Fn(t) _

The functions Fn and µ are multiplicative, therefore the same is true for the convolution µ * F. (see Theorem 2.2). (e) By the definition (3.2) of the RAMANLUAN sum the proof of the orthogonality relations (3.3) is reduced to an application of the corresponding relations for the exponential functions. More explicitely:

r

r

cd(m) m=1

I

ct(m) = r

r

e(a m) t e(b m)

a>'

gcd(a,d)=1

gcd(b,t) =1

I a sd

bst

gcd(b,t)=1

gcd(a,d)=1 0 =

,

if

r

msr

e(d m)

e(t m)

d $ t, or a+b * 0 mod d,

la-d

lbsd

gcd(a,d)=1

1

= p(d) otherwise.

gcd(b,d)=t

a+b= 0 mod d The reason for the last equality-sign is that for every a there is exactly one b, satisfying a + b = 0 mod d. 11

In Chapter IV we shall need some special values of cr(n). If the index r is a prime power pk, then, as is easily verified, -

pk

(3.4)

cpJn)=

- pk-1 0

,

p k-1

if p In, if pk-111n, if pk-1 t n.

Tools from Number Theory

18

Figures 1.6 and 1.7 illustrate the periodic behaviour of RAMANUJAN 30 and sums rather instructively. The functions c r with index r r = 210, resprectively, are plotted in the range 1 s n s 299. 10

so

loo

lso

200

300

Figure 1.6: RAMANUJAN sum c30 In the range 1 s n s 299 10

too

200

300

Figure 1.7 RAMANLIJAN sum c210 In the range 1 s n s 299

Other examples of r-even functions are 1, if gcd(n,r) = d, gd: n H 1

0, otherwise.

The functions gd, where dir, as well as the RAMANWAN sums cd,

1.3. Periodic Functions, Even Functions, Ramanujan Sums

where

19

dlr, form a basis of the C-vector-space of r-even functions

(this space is of dimension t(r)). This is obvious for the functions gd, and for the RAMANUJAN sums the assertion easily follows from the orthogonality relations. a The KRONECKER-LEGENDRE symbol P is equal to zero if pla; otherwise, if p]' a, it is equal to 1 or -1 If a is a quadratic residue

[resp. non-residue] modulo the prime p. (p) is a completely multiplicative, p-periodic function (considered as a function of the "nominator" a). For a thorough investigation of the LEGENDRE symbol as a function of its "denominator" p, see, for example, H. HASSE [1964]. This function a

Generally, given a character X of the group

(

7L/m7L )

x

of residue-

classes prime to m, in other words, given a group-homomorphism X : ( Z/mZ )x --) ( C,

.

),

IX(n)I = 1,

we obtain a completely multiplicative, m-periodic function X

:

IN -*{ z E C, Izi = 1 or z = 0 },

defined by X(n) = X(n mod m) If gcd(n,m) = 1, and X(n) = 0 otherwise.

1.4. THE TURAN-KUBILIUS INEQUALITY

An additive function w: IN - C is called strongly additive if the values of w at prime-powers are restricted by the condition w(pk) = w(p), if k = 1,

2, ...

.

In 1934, Paul TURAN [1934] discovered the following inequality for the strongly additive function n H ca(n), the number of prime divisors of n: (4.1)

1 ((j(n) - loglog x )2 s c

nsx

x

loglog x

with some constant c. P. TURAN used this result to reprove HARDY

Tools from Number Theory

20

and RAMANL[IAN's theorem [1917] that ro(n) has normal order loglog n. Inequality (4.1) was generalized by J. KuBILIUS [1964] to additive functions, and later "dualized" by P. D. T. A. ELLIOTT [1979]. If w is strongly additive, then

Z w(n) = Z

Z w(p) = pox Z w(p)

[x/Pl,

nsx pin

nsx

and so w(n) is, on average, heuristically approximate to 2: p-<X p w(p), The so-called TUPAN -KuBILIUS inequality gives an estimate for the 1

difference of the values of the function minus the "expectation":

w(n) - Y_ p_'-w(p) psx

in mean square. its general form the TURAN-KUBILIl1S inequality has often been applied to the study of arithmetical functions. We use this inequality in Chapter VII in order to approximate functions in 11 by even functions In

and to outline criteria for additive

and multiplicative functions to

belong to 21q

For an arithmetical function w (and x > 0) we define the expressions A(x) =

(4.2)

Z P -k. w( pk),

P SX

E(x) =

(4.3)

P

and

P`sx

p

k' Iw(Pk)I2.

Theorem 4.1 (Tur'n-Kubilius inequality). There exist constants C1, C2 with the property that for every x2 2 and for any additive function w the inequalities (4.5)

xI

nsx

I w(n)

- A(x) 12 s CI D2(x)

and (4.6)

x

> I w(n) - E(x) 12 s C,-D2(X)

nsx

are true. In fact, it is possible to have C, = 30, C2 = 20.

14 The Turdn-Kubilius Inequality

21

Remark. If w is strongly additive, then the CAUCHY-SCHWARZ inequality gives

A(x) -Z p p-1'w(p)

I

I

s

s p.

P-1'Iw(P)I2)1

psx

P-2'Iw(P)I

p-k,I w(Pk )I 5 2 2:

k22

P:rx

(

2

psx

5 (1 psx

p-3)

P-1'Iw(p)I2).,

and

D2(x) s 2 E psx Therefore, from (4.5) we deduce (4.7)

x-1

I

nsx

w(n) -

P-1. w(p) I2

psx

s 2 x-I Ix

( 4 CI + 2

s

for

I w(n) - A(x) 12 + 2 I A(x)

-

psx

1 psx p-1 . Iw(P)I2

every strongly additive function w. Note that the constants are far

from being the best possible.

Proof of Theorem 4.1. Inequality (4.5) is a consequence of (4.6). By appropriate application of the CAUCHY-SCHWARZ Inequality we obtain E Iw(n) - E(x)I)2 x Iw(n) - E(x)I2 s nsX

nSX

and

s x-I

E Iw(n)-E( x)I2 + 2x-1 E Iw(n)-E(x)I

nsx

nsx

+

(

Ep

p`sX

p-k-I.I w(Pk )I)2 5 (C2 + 2 C2' +

-k-1 .I w(Pk)

1

I

D2(X).

)

We follow the proof given in ELLIOTT [19791, p. 148. There is another proof, due to ELLIOTT 119701, which uses the "large sieve". First, the assertion for complex-valued functions is reduced to the corresponding assertion for real-valued functions in an obvious manner, and then the

assertion for these functions is reduced to a problem concerning nonnegative functions.

(i) Assume initially that w is real-valued and non-negative. Then S=

nsN

(w(n)-E(N))2 = 2]

nsN

w2(n) - 2 E(N)

nsN

w(n) + N

E2(N)

Tools from Number Theory

22

= S1 - 2 S2 + N

E2(N)

say.

,

First, S1 =

GN psN

w2(n) _ 21nsN Xp`IIn w(Pk)

w2(P )

'

1

nsN

+

ZgQIIn W(qQ)

w(p k)

plq'esN p*q

p` Iin

w(q- )

.

a(N),

where, for distinct primes p and q, #(N)

I

Zn,N, p`IIn, gEIIn

counts the number of integers, which are exactly divisible by the prime powers pk, q1. Then

+

[ N/ p kqe+l

Pk+1'qt

u(N) = [N/pk,q.e ] - C N/

1

[ N/ pk+1,qt+1

(4.8)

_

N/pk.q'

-

1

1

P

-

)

q

+ 20,

where 101 s 1, and therefore

S1 s N

D2(N) + N ' E2(N) + 2

P*q

w(pk) ' w(qt)

Second,

EnsN'p`IIn w(pk) = Gp,sN w(pk) .

Lp-k.N] (

-

[p-k-1

N]

N - Ep.SN w(pk).

p-k, Z

Putting these estimates together, the term E2(N) cancels. Application of the CAUCHY-SCHWARZ inequality gives N-1 S s D2(N) +

w(p k) E p`gEsN, p*q

s It + 2'N-1 ( p'gQsN, P*q pk

.

w(qt) + 2-N-'- E(N) . Z

p`sN

q'e+ p`sN

pk E p kD2(N)

Some standard estimates complete the first part of the proof:

k p*q p

qe)S

w(Pk)

2

-r2-,

j.4. The Turn-Kubilius Inequality

Y-p.SN p

ks

23

41

ZpwSN pk S N'

nsN

n 1 S log N,

log-1(N) N a 2.

1 S 8 N2

The last estimate uses ZpSN log p s 2-log 2' N (for N 2 2 ). This implies PIN I + ZPwsN,kk2 I

S

+ ZpsN ( log W) -1

log p +

21ks log N/ log 2 psN,.` 1 s Ni + (2/log N) ' 2 log 2 ' N + ( log N/log 2 ) ' N21 S 8 N ' log-1(N). S

ps,/N

1

Thus the method gives the constant 1 + 2 ' 2 + 2'i S 10. Due to part (ii) this implies C2 = 20.

If w is

real-valued, define additive functions f+, f , where f+(Pk) = max (0, w(Pk) ), f (Pk) = - min (0, w(Pk) ). Define E+(N), ..., (ii)

D (N) in the same manner as E(N) and D(N), but now using the functions f+ and C. Then I

w(n) - E(N) 12 S 2 ( If +(N) -

E+(N) 12

+ If (N) - E (N)12 ),

and utilizing the relation D+(N)2 + D -(N)2 = D2(N), we obtain the result for real-valued functions, unfortunately with an additional factor of 2.

(Iii) The case of complex-valued functions is reduced to the real-valued case in the usual way by decomposing f into its real and imaginary part.

Next, we are going to "dualize" the TUBA N-KUBILIUS Inequalities. Consider the complex vector-space CM of all vectors 2 = (z1, z2, ..., zM) with Euclidean norm (4.9)

11211=(

and the usual inner product <

,

E Izm 12 msM .

>.

For linear maps L: CM -4

CN

the "operator-norm" (4.10)

11

L II

=

11;2 11

=1

II

L(a)

II

is used. If, with respect to the canonical basis, the matrix C = (cm,n ),

Tools from Number Theory

24

1 s m s M, 1 s n s N, is associated with L, then the adjoint operator L* ( defined by < Lx, ti > _ < x, L*14 > ) is connected to the matrix Ct. Because of

II L II

=

II L

II we obtain the following result.

Theorem 4.2 (ELLIOTT's Dualization Principle). Let cm,n ' m =

1,

""

M,

n = 1, ..., N, be complex numbers, and let c > 0 be given. Then the inequality 2:.:,N I Zm:M Cm,nZm 12 5 C

(4.11)

ImsM IZm I2

is valid for all 2 e CM if and only if the "dual Inequality" (4.12)

Y-msM 1

2:nsN Cm,n Wn 12 5 C

2:nsN 1 Wn 12

is true for every to e CN.

Applying this principle to Theorem 4.1 yields the following theorem.

Theorem 4.3. For all x2:2 and for all complex sequences (w n ) the following Inequalities are valid: k'Ip

(4.5')

p°sx

(4.6'' )

GP

P

kP

x

pYsx

X nsx,p"IIn

1

- x nsx Z

wn

wn

W'(1-p1)' nsx nsx,p"IIn n x

12

C

l

2: x nsx

wn 2 5 C 2

1wn12,

1' nsxZ x

I W 12.

Further examples of applications of the dualization principle are given in the exercises, p.41. Finally, there is the following generalization of Theorem 4.1 to higher powers: Theorem 4.4 (TURAN-KUBILIUS-ELLIOTT Inequality). Given q

z

0,

there is a constant c > 0, so that the Inequalities x1

'

nsx

I

w(n) - A(x) I`' S c x1

I

nsx

w(n)

'

(Dh(x) +px I

P-k.

I

W(Pk)I9 ), If q > 2,

A(x) I4 s c' D9(x), if O s g s 2,

are valid for every additive function w and every x >-2.

The special case where q = 2 is Theorem 4.1 (only the numerical value of the constant c is not specified). We do not use this generalization,

and so we do not prove it, but, rather refer the reader to P. D. T. A. ELLIOTT 11980c].

I .S. Generating Functions, Dirichlet Series

25

I.S. GENERATING FUNCTIONS, DIRICHLET SERIES

The study of meromorphic functions near their singularities leads to arithmetical insight. In order to obtain meromorphic functions associated with arithmetical functions, different kinds of generating functions are used which are often treated purely formal; among the best known are LAMBERT series, generating power series and DIRICHLET series. (a) LAMBERT series: associate with a given arithmetical function

f; N -- C the infinite series L(f,z) = Znal

f(n)-z° (1-z°)-1.

Then, assuming absolute convergence in the [open] unit disc

B(0,1) = (z E C; Izi < 1),

the series L(f,z) can be transformed into L(f,z) _ Zn21 f(n). Zk20 zn(l+k)

rz1

z"

(

1

=

rx1 zr- Zdir f(d)

* f )(r).

Examples. In Izi < 1 the LAMBERT series L(1, z) _ n21 t(n) z° , since 1 * I = t , and L(X,z) _ n_t z°, where A is the completely multiplicative function taking the value -1 at every prime p. It is easily checked 2

that 1 * X = 1sq' the characteristic function of the set sq of squares.

If some suitable condition restricts the growth of the arithmetical function f, then the LAMBERT series L(f,z) is holomorphic in B(0,1). In = 1. This is true

general, there will be a singularity of L(f,z) at z

when the convolution 1 * f is non-negative and infinitely many of the values (I * f)(r) are non-zero, for example. Conclusions about the behaviour of the coefficients are often possible with the aid of Tauberlan Theorems; some of these theorems, important in number theory, are summarized in the Appendix (A.4).

Tools from Number Theory

26

(b) Generating power series.

We associate with the function f :IN0 - C the power series

9'(f,z) = nz0 f(n)

(5.1)

Zn.

If the function f is not too large, the power series (5.1) will converge in the complex unit circle B(0,1) = { z E C, Izi < I }. In order to obtain arithmetical conclusions, the most interesting singularity is generally the point z = (if this point is a singularity. This is certainly true, If f is non-negative and if infinitely many values of f are non-zero), However, in the case of the partition function n H p(n), for example, 1

with generating power series (S.2)

n=0

p(n) . zn = k=1 TI ( 1 - zk)-1

there are many other singularitites which have to be investigated if better estimates of the remainder term are desired. The method to be used is the analytic HARDY- LITTLEWO OD- RA MANUJAN circle method;

the coefficients p(n) of the power series (5.2) are expressed through a contour-integral, the main contribution to this integral being from small arcs of the integration path near the singularities of the function on the right-hand-side of (5.2). A useful device is outlined in HALL-TENENBAUM 11988]. If f Z 0, and 9D(f,z) converges in some interval I of the real axis including the point 1, then, obviously, for any N > 0, (5.3')

nsN nxN

f(n) s

f(n)s

f(N) s

inf x N . P(f,x),

O<xs1

inf x N x21,xcI

inf

x>O,xcl

x-N

9'(f,x),

P(f,x).

For example, using f(n) = (n!)-1, the last equation gives (N !)-1 s (e/N )N,

a rather good lower estimate of (N factorial). Another application of this principle is also taken from HALL-TENENBAUM 11988], section O.S. If E is a set of primes with least element

po(E), E(x) = Xpsx,pcE p-1, and Q(n,E) the total number of prime-

I .S. Generating Functions, Dirichlet Series

27

of n [counted with multiplicity] which lie inside E, then, for

divisors 0c< y < po(E ), the following Inequality is a consequence of II. Theorem 3.2: yf](n,E) << (x/ log

G

x)

nsx

TI psx

-L-

P

TI (I psx

)

p-1 \-1

pIE

pcE

< x e(y-')E(x)

J

Uniformly in 0 s k s (po(E) - c) E(x), taking y = k / E(x), (5.4) yields

#(ns x, fl(n,E )= k } s of

y-k.

YO(n,E)

2x

E k(x) k!

x, e- E(x)

<< x- e-E(x),ek (E(x)/k)k.

r k-.

As an example of the use of power series in additive number theory, we mention, for a given sequence A = (aV) of integers 0 s a1 s a2 s ... (where at most O(m) repetitions for am are allowed) the power series g(Z)=

n

Zg

this series forms the basis of the proof of the famous ERDbs-FuCHS Theorem. Abbreviating the number of representations of n in the form n = aV + a µ to Rn, this theorem states that, as N -) -, the relation n=o

R. =

log-N )

o( Na

cannot hold. The proof exploits the generating power series ZN=o (

n=0 Rn )

,

ZN = ( 1 - Z )-I

.

g2(Z)

For details see H. HALBERSTAM and K. F. ROTH, Sequences [1966], II. § 4.

(c) DIRICHLET series.

The most suitable of the generating functions for the investigation of multiplicative arithmetical functions are generating DIRICHLET series,

£(f,s) = gnat f(n)

(5.5)

.

n-s,

which are defined and holomorphic in some half-plane o = Re s > 60, if f does not increase too quickly. Multiplicativity of f Implies a product representation for £(f,s) by the following lemma. Lemma S.I.

If f is a multiplicative function for which 2:n2t

absolutely convergent, then there is a product representation

f(n) is

Tools from Number Theory

28

nE1 f(n) = IT (1 + f(p) + f(p2) + ... )'

(5.6)

where all the series 1 + f(p) + f(p2) + ... and the product itself are absolutely convergent. Lemma 5.1 is well known (see, for example, HARDY-WRIGHT [1956],

Theorem 286); its proof is left as Exercise 19. Applying the lemma to the DIRICHLET series Vf,s), we obtain the following corollary.

Corollary 5.2. If f is multiplicative, then in the region of absolute convergence of the DIRICHLET series B(f,s), the product representation (5.7)

Z f(n)-n-' = II ( p

nzl

1

f(p2),p-2s

+

+

...

is valid.

One of the simplest DIRICHLET series is the RIEMANN zeta-function (5.8)

c(s) = D(1,s)

nal

n

= IfP ( 1+p-s+p-2s+...

s

=r

(i-p-sY1

which is absolutely convergent in Re s > 1. This product representation indicates some connection with the theory of prime numbers. EULER'S summation formula (see Theorem 1.2), applied to In21 n-5, gives (5.9)

((s) _ (s - 1) 1

+

2 - s fi Bo(u) u cs+l> du.

The integral defines a holomorphic function in Re s > 0, and so formula (5.9) provides an analytic continuation of C(s) into the half-plane Re s > 0, showing that c(s) has a simple pole at s = 1 with residue 1. Further integrations by parts of the integral occurring in (5.9) give the

analytic continuation of C(s) into the whole complex plane; which can be achieved in one stroke by the functional equation

(5.10)

c(s) = 2s

r<9-1

sin(

its

) F(I-s)

c(1-s)

of the RIEMANN zeta-function.

The pointwise product of two DIRICHLET series is (in the region of absolute convergence of both DIRICHLET series) given by

I .S. Generating Functions, Dirichlet Series

£(f,s)

D(g,s) =

f(d)

nz1 dTn

'

29

g(n/d) '

n-s

= D(f*g,s),

and so the pointwise product of the DIRICHLET series corresponds to the convolution product of arithmetical functions. Thus, consideration of DIRICHLET series is useful for multiplicative functions and also in connection with convolutions of arithmetical functions.

Noting this remark and the convolution identities t

=

1*1,

I* V

=

s,

(p = id* v, one obtains formulae for some generating DIRICHLET series: 2 2(s)

n=1 (5.11)

V not ti(n) l

p(n)

n-s '

n-s

,

Re s

= -1(s), Re s >

1,

Re s > 2.

=

Many other formulae of this type are given in HARDY-WRIGHT [1956], and a general theory of "Zeta-Formulae" is developed in J. KNOPFMACHER's book [1975] on abstract analytic number theory.

The connection of the RIEMANN zeta-function with the theory of prime numbers arises from the generating DIRICHLET series

Zn

(5.12)

1

A(n)

n-=

Re s > 1,

where the VON MANGOLDT function A is given by (5.13)

log p

A(n) = l 0

5.000

,

if n is a power pk of the prime p, otherwise.

10.000

15.000

F i g u r e I.8. Primes in intervals of length 100

20 000

Tools from Number Theory

30

The number rt(x) of primes in the interval [1,x] behaves rather erratic locally. This is illustrated in Figure 1.8 on the foregoing page, giving the number of primes in intervals of length one hundred, from k s 199. The first interval contains twenty-five primes, the next one twenty-one, etc. , but there is also an interval containing only five primes.

The problem of obtaining an asymptotic formula for the number of primes up to x , rc(x) = > psx

(5.14)

1

,

is equivalent (via partial summation) to a suitable approximation to

,9(x) = I log P psx

(5.15)

or to (5.16)

4(x) = I A(n), nsx

via the easily verified relation [use the fact that higher powers are rare]

O(x) = E Mn) + of x' (log x)2 nsx

.

The function 4(x) = 1nsx A(n) has an integral representation (by a complex inversion formula), (5.17)

4(x) = (27d)-1

.

('c+i. C-1-

(-

s-1 x9 ds,

where c > 1. The "Method of Complex Integration" allows approximation

of the integral in (5.17) by shifting the path of integration to the left. The pole at s = 1 gives the main term x. Further contributions to the asymptotic

(- '(s)/i (s))

formula s-I xs,

follow

from

poles

the integrand which are caused by zeros of the RIEMANN

zeta-function in 0 < Re s <

the

of

1.

For the [lengthy] details of this method see, for example, PRACHAR [19571, DAVENPORT [1967], SCHWARZ 119691, HUXLEY [1972], Ivtc

[1985], TITCHMARSH [1951], or other monographs on the theory of primes.

1.6 Some Results on Prime Numbers

31

1.6. SOME RESULTS ON PRIME NUMBERS

It will frequently be necessary to use asymptotic formulae or estimates

for sums or products running over primes. We cannot prove these [standard] results, but quote some of them for easy reference. The method described at the end of the preceding section, In combination with some deeper knowledge on the distribution of the zeros of the zeta-function In the critical strip z < Re s < 1, gives the Prime Number Theorem:

Theorem 6.1. For x - co, with some positive constant y, the following asymptotic formulae hold: (6.1)

4(x) = x + 0 (

y (log x)2) ),

(6.2)

9(x) = x + O (

Y (log x)2

(6.3)

7t(x) = Ii x + C'7 (

)

),

y (log x)2

where li x Is an abbreviation for the so-called Integral-logarithm

Ii x = li e + f e" (log u)-1 du, Ii e = 1.895 117 8...

(6.4)

Some [rounded] values of it(x), x

Ii

x, and x/log x are given in Table I.I.

100

1000

n(x)

25

168

1229

9592

78498

Ii x x/log x

30

178

1246

9630

78628

21.7

144.8

1086

8686

72382

105

10 4

10 6

Table 1.1

The function 11(x) is connected with the Exponential-Integral Function Ei(x) by the formula 11(x) = Ei(log x), and may be calculated from the series development

li x = e + loglog x +

1sn<m

( log x )n / (n

n!

), for x >

1.

Tools from Number Theory

32

C = 0.577 21S 664 90. ... is EULER's constant. Roughly speaking, the inte-

gral-logarithm behaves as x/log x, so that xlim m li x / ( x/ log x) = I. It is possible to deduce from (6.4) an asymptotic development of li X

by partial integrations, for example (with three main terms on the right-hand side) :

x,log-2x + 2,x,log-3x + O(

li x =

A. SELBERG, P. ERDOS showed in 1948 (independently) that the prime

number theorem may also be obtained by "elementary methods", We

present one of these elementary proofs, due to H. DABOUSSI,

in

11.§9. Rather simple, elementary methods lead to the estimates given below, which are frequently required. For proofs, see, for example, PRACI-IAR 119571 or SCHWARZ (19691. Theorem 6.2. There are constants 0 < c1 < c1

(

1

< c2 such that

x/log x) < n(x) < c2

(6.5)

c1

x < ,4(x) < c2

c1

x < fi(x) < c2

( x/log x),

'

X,

X.

Furthermore, as x - oo, 1psx

(6.6)

p = log x + 0(1),

zpsx p-1 = loglog x + Y2 + 0(1),

(6.7)

and

fT (1 -

(6.8)

Ps x

p-1) = e-o'( log x)-'-(l + o(l) ).

The remainder terms in these formulae may be improved by using the prime number theorem. For example, with the "standard remainder term" 0( x' exp(-'y log x )) of the prime number theorem, we obtain (6.9) (6.10)

21

psx

psx

p-1 -log p = log x + y1 + c exp( -Y

log x

P-1 = loglog x + Y2 + C7( exp( -y

log x )),

1.6. Some Results on Prime Numbers

33

and

P-1) = e-1°, (log x) -1 lI (6.11)P rT (1 -

+

O { x exp {-Y

log x } } ),

(kpk)-t and i° = 0.577 215 664 901... is EULER's where Y2 = ti° - 2: , 2: k22 constant (see, for example, PRACHAR [1957 ] ), and x tends to infinity.

Many estimates and Inequalities of this nature, with explicit constants and often very deep, are given in ROSSER & SCHOENFELD [1962].

µ(n) of the MOBIUS-function

The "mean-value" M(µ) = limx-), .0 is zero. More exactly, n

(6.12)

N

µ(n) = C7 N exp{-Y log N

The function N H NI nN µ(n) is plotted below for N = 2, 4,

...

.

0.5

0.1

0

- 0.5

40

200

400

600

8 00

1000

1200

F i g u r e I.9 Sum over the Moblus function

VON STERNECK's conjecture, supported by Figure I.9, states that CAI-1 X nsN

µ(n)I s z, if N > 200.

This conjecture is not true (G. NEUBAUER [1963] ), and the weaker MERTENS conjecture, where a is replaced with 1, is also not true (A. M. ODLYZKO & H. J. J. TE RIELE [1985]. See also TE RIELE [1985]. JURKAT& PEYERIMHOFF proved a weaker result in 1976).

The S60 primes in the interval [2, 4057] are given in Table I.2.

Tools from Number Theory

34

Figure I. 10 represents the primes between 1 and 10.000. Some explanation

is necessary. The small rectangles mark the integers, beginning with 1

in the bottom line up to 100, from 101 to 200 in the second from bottom line and so on. A dark rectangle indicates that the integer represented by this rectangle is a prime. Note: the column with columnindex 10, 20, ... Is to the left of the vertical line 1. For example, the top line contains the nine prime numbers 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, and 9973.

Table of Prime Numbers 2

31 73

3 37 79 131 181 239 293

127 179 233 283 353 419 467 547 607 661 739 811 877 947 1019 1087 1153 1229 1297 1381 1453 1523 1597 1663 1741 1823 1901 1993 2063 2131 2221 2293 2371 2437 2539 2621 2689 2749 2833 2909 3001 3083 3187 3259 3343 3433 3517 3581 3659 3733 3823

359 421 479 557 613 673 743 821 881 953 1021 1091 1163 1231 1301 1399 1459 1531 1601 1667 1747 1831 1907 1997 2069 2137 2237 2297 2377 2441 2543 2633 2693 2753 2837 2917 3011 3089 3191 3271 3347 3449 3527 3583 3671 3739 3833

3911 4001

3917 4003

5

41 83 137 191 241 307 367 431 487 563 617 677 751 823 883 967

7 43 89 139 193 251 311 373 433 491 569 619 683 757 827

19

23

29

61

1487 1559 1619 1699 1783 1871 1949 2017 2089 2161 2267 2339 2393 2473 2579 2663 2713 2791 2861 2957 3041 3137 3221 3313 3373 3467 3541 3617 3657 3779 3863

17 59 183 157 211 269 331 389 449 509 587 643 789 773 853 919 991 1051 1117 1201 1279 1327 1433 1489 1567 1621 1709 1787 1873 1951 2027 2099 2179 2269 2341 2399 2477 2591 2671 2719 2797 2879 2963 3049 3163 3229 3319 3389 3469 3547 3623 3701 3793 3877

107 163 223 271 337 397 457 521 593 647 719 787 857 929 997 1061 1123 1213 1283 1361 1439 1493 1571 1627 1721 1789 1877 1973 2029 2111 2203 2273 2347 2411 2503 2593 2677 2729 2801 2887 2969 3061 3167 3251 3323 3391 3491 3557 3631 3709 3797 3881

67 109 167 227 277 347 401 461 523 599 653 727 797 859 937 1809 1063 1129 1217 1289 1367 1447 1499 1579 1637 1723 1881 1879 1979 2039 2113 2207 2281 2351 2417 2521 2609 2683 2731 2803 2897 2971 3067 3169 3253 3329 3407 3499 3559 3637 3719 3803 3889

113 173 229 281 349 409 463 541 601 659 733 809 863 941 1013 1069 1151 1223 1291 1373 1451 1511 1583 1657 1733 1811 1889 1987 2053 2129 2213 2287 2357 2423 2531 2617 2687 2741 2819 2903 2999 3079 3181 3257 3331 3413 3511 3571 3643 3727 3821 3907

3931 4021

3943 4027

3947 4049

3967 4051

3989 4057

11

13

47 97 149 197 257 313 379 439 499 571 631 691 761 829 907 977

53 101 151 199 263 317 383 443 503 577 641 701 769 839 911 983 1049 1109 1193 1277 1321 1429

1031 1093 1171 1237 1303 1409 1471 1543 1607 1669 1753 1847 1913 1999 2081 2141 2239 2309 2381 2447 2549 2647 2699 2767 2843 2927 3019 3109 3203 3299 3359 3457 3529 3593 3673 3761 3847

887 971 1033 1097 1181 1249 1307 1423 1481 1549 1609 1693 1759 1861 1931 2003 2083 2143 2243 2311 2383 2459 2551 2657 2707 2777 2851 2939 3023 3119 3209 3301 3361 3461 3533 3607 3677 3767 3851

1039 1103 1187 1259 1319 1427 1483 1553 1613 1697 1777 1867 1933 2011 2087 2153 2251 2333 2389 2467 2557 2659 2711 2789 2857 2953 3037 3121 3217 3307 3371 3463 3539 3613 3691 3769 3853

3919 4007

3923 4813

3929 4019

T a b 1 e 1.2. Prime Numbers below 40S8

71

17. Characters, L-Functions, Primes in Arithmetic Progressions 10

20

30

so

70

90

20

30

so

70

90

3S

100

F I g u r e I.1O Characteristic Function of the Primes below lo.ooo

1.7. CHARACTERS, L-FUNCTIONS, PRIMES IN ARITHMETIC PROGRESSIONS

For a treatment of primes in arithmetic progressions ( primes p in residue-classes p = a mod q, where gcd(a,q) = t ) it is necessary to have functions that single out the elements of one residue-class. Such functions, which are, in addition, multiplicative and periodic, are the DIRICHLET characters, defined on IN or Z. Characters will be introduced

in a more general setting: we assume that g is a topological group, which, in addition, is also abelian.

A character y on § is a continuous homomorphism from circle group (7.1)

T = { z E C ;

IzI

=

1

)

9

into the

Tools from Number Theory

36

by multiplication,

X:(§,')-(T,') By pointwise multiplication the characters on § again form an abelian group, I. e. the character group (y:

(7.2) This group

- T, y continuous homomorphism}.

can be converted to a topological group in the following

manner: a basis of neighbourhoods of the unit element e of § consists of the sets

U(e,K)={XE§ , IX(x)-1I <sforall xInK}, where s is any positive real number, and K c § any compact set in As mentioned already in section 3, a character X defined on the group ( 7L/m7L )x = {a mod m, gcd(a,m) = 11

of residue-classes a mod m, prime to m, with discrete topology, in other words a group-homomorphism X:

x

( Z/mZ ) -> ( C,

X(n)I = 1,

induces a completely multiplicative, m-periodic function (7.3)

X

: N -) { z

E

C,

IzI

=

1

or z = 0 },

defined by X(n) = x( n mod m ), if gcd(n,m) = 1, and x(n) = 0 otherwise.

The unit element of the character group induces the so-called principal character Xo mod m, with values X0 (n) = 1, if gcd(n,m) = 1, X0 (n) = 0 x otherwise. The character group of ( 7L/m7L) has p(m) elements. These functions (7.3), called DIRICHLET characters, allow the construction of DIRICHLET L-functions

L(s,X) =n=1 E X(n)'n

s

= IT (1 + x(P) p

P-9

+ x(P2)

P-2s

+

...

(7.4)

= I-T {1 - X(P).p-,

1.

The series and products in (7.4) are absolutely convergent in Re s >

1.

Moreover the series Zn 1 is conditionally convergent in Re s > 0 if y is not the principal character. If X is the principal character Xo, then

j.7. Characters, L-Functions, Primes in Arithmetic Progressions

L(s, Xo) = 11

p.i'm

1 - p-S )-,

= pIm n (1 -

p-S l

l

37

c(s)

DIRICHLET characters X satisfy (like characters in locally compact

topological abelian groups In general, where summation is replaced by integration with respect to the HAAR measure on ;°) the orthogonality relations:

If a runs through a full set of representatives mod m ( for example, a = 1, 2, ..., m ), then X(a) =

a mod m

(7.6)

( cp(m),

If X is the principal character, otherwise.

0

If y runs through all the p(m) DIRICHLET characters mod m, then X(a) - J rp(m), if a = I mod m,

(7.7)

otherwise.

0

Corollary (Orthogonality Relations for DIRICHLET Characters). (7.8)

a mod m

X1 (a) '

X2(a) _

rp(m), if Xl = XZ, 0

otherwise,

and

9(m), if at = a2 mod m and I X(a1) x

.

gcd(ala2, m

X(a2) _ 0

) = 1,

otherwise.

These relations allow specific residue-classes mod m to be singled out: If f is an arithmetical function and gcd(a,m) = 1, and if n', are positive integers, then (7.10)

tsJ, nL=a mod m f(n) = 9(m) 1

x

X(a)

LSJ

f(n

I.

= 1, ..., J,

I.

Since

- L'(s,X)/L(s,X) = In= X(n)

A(n)

.

n-S

one finds results on primes in arithmetic progressions in the same way as is possible for ordinary primes (for example, using the method of "complex integration"). DIRICHLET L-functions have properties similar to those of the RIEMANN zeta-function, and so, using the functions

Tools from Number Theory

38

TE(x;a,q) = Ipsx, p=a mod q 8(x;a,q) =

(7.11)

1,

psx, p@a mod q log P,

(x;a,q) = nsx,

mod q A(n)

one obtains the following theorem. Theorem 7.1. If gcd(a,q) = 1, then, with some positive constant y, depending on a and q, the following asymptotic formulae hold : , 1

rp(q)

1q

(7.12)

cp (

)

1

x + C (x'exp( - y (log x) 2

) ),

,

x + O ( x'exp( - i (log x)2) ), Ii x + 0(x'exp( - y (log x)2 ) ),

W(q)

with the Integral-logarithm Ii x =Ii e + fe

(log u)-1 du.

It is sometimes important to have uniform estimates on n(x;a,q) in

I s a s q, with q restricted to some range, depending on x. An Important result of this kind is provided by the following theorem. Theorem 7.2 (Prime Number Theorem of PAGE-S!EGEL-WALFISZ). If I s a s q, if gcd(a,q) = 1, and if I s q s ( log x )A with some fixed constant A, then, as x tends to infinity, the asymptotic formula (7.13)

Tt(x;a,q

) = w(q) Ii x + 0A (

Y (log x)2} )

holds uniformly in a and q. As indicated, the constant Implicit In the 0-symbol may depend on A. For a proof see, for example, PRACHAR [1957] or ESTERMANN 119S21.

For some applications the range of admissible values for q in (7.13) is not sufficient, for example when consideration of larger values of q is unavoidable; this occurs in problems from the additive theory of numbers. The sieve method (V. BRUN, A. SELBERG) or the "large sieve" easily gives the upper estimate (7.14)

n(x;a,q) < Y '

x/(

p(q)log(x/q)

I.S. Exercises

39

With some constant y (which, in fact, may be taken to be 2, as long

as q

<

for some positive constant 8), See, for example, MONT-

GOMERY [1971], HALBERSTAM & RICHERT 119741, or SCHWARZ 11974].

Another deep and extremely useful result Is the prime number theorem of B. BOMBIERI and A. I. VINOGRADOV, which says that on average (7.13) is

true [with a better remainder term] for a much larger range of values of q. Theorem 7.3. (BOMBIERI-VINOGRADOV's Prime Number Theorem). For

any positive constant A there Is a [positive] constant B, for which the estimate E

max

max I n(y;q,a) q:9 Z1og-Bx y Sx a mod q, gcd(a,q)=1

11 y

9(q)I

_

OA(Xlog Ax

Is true. For a proof, see, for example, DAVENPORT [1967] or HuXLEY 11972].

I.B. EXERCISES

Remark. Many similar exercises may be found, for example In APOSTOL 119761.

Deduce higher EULER summation formulae such as

1) Y-

a<nsx

g(n) = f xg(u)du + f a

g(x)-Bo(x)- 2 g(a)+

a

where a is an integer, and B1(x) = f XB0 (u)du = a

2)

Prove (by partial summation) for every real s > I and for x > 2

nsx

+ s JiLL-(s+1),B (u) du I s x s. 0

Prove, for s > 0 and x 2 1,

Z.,x

ns -

(xs+1

- 1)

(s+l)-1

I

s x'.

1

T.

Tools from Number Theory

40

3) Exhibit infinitely many functions, linearly independent in the vectorspace C N. Prove orthogonality relations for the functions gd, dir, defined in 1.2.

4 ) Denote by µr(n) the characteristic function of the set of r-free integers (r = 2, 3, ... ), so that iir(n) = 0, if there is some prime for which pr divides n, and µr(n) = I otherwise.

Prove: The function µr is multiplicative and ilr(n) = Ed,in µ(d).

5) Denote by p(n) the number of solutions d mod n of the congruence f(d) = 0 mod n, where f(X) is a non-constant polynomial with integer coefficients. Prove that p is a multiplicative function. µ log, A * MANGOLDT'S function ( see 1.6 ).

6) Prove I * µ2 = 2", and A * µ

7) Show that

= q

(n)

din

U(

1

= log, where A Is VON

)

gyp (d)

8) The LIOUVILLE function A: n H (-1)0(n) has the convolution inverse a-1(*) = more generally a completely multiplicative function h The function 61: n H zdin d has convolution inverse h-l(*) = has the inverse It * (V-id).

9) If ft is completely multiplicative, and f2 > 0 is multiplicative and integer-valued, then ft^f2 Is multiplicative. 10)

If f

is

a

multiplicative solution of the functional

equation

fZ = 2" * f, then f is integer-valued. 11) Prove or disprove the following:

(a) If f is r-periodic, f(1) = 1, then f -l(*) is r-periodic. (b) If r is r-even, f(1) = 1, then f-t(*) is r-even.

(c) Every r-periodic function is s-even for some positive integer s. (d) If f is strongly multiplicative, then f-1(*) is 2-multiplicative. (e) If f is 2-multiplicative, then f 1(*) is strongly multiplicative. 12) The RAMANUJAN sum n ycr(n) is multiplicative if and only if µ(r) = 1 [so that r = 1 or r is a product of an even number of different primes]. 13) (HOLDER 1936). Put n' = r/gcd(n,r). Then cr(n) = µ(n')(y(r)/cp(n')).

I.S. Exercises

41

1925). Denote by f(n,r) the number of solutions of the linear congruence

14) (RADEMACHER

in vectors ( xp mod r gcd( xp,r ) = 1. Prove: (a) f(n, r1 r2) = f(n, r1) then (b) If

.

sass

)1

with the additional condition

f(n, r2), if gcd(r1, r2) = 1.

f(n, Pk) = pks-k-s .

(P-i) s + (-i )s-1 },

{

and for pIn pks-k-s

f(n,pk) =

'(P-1)

{ (p-1) s-1 + (-1)s }.

15) The vector-space CN with multiplication

f i g: n H Zdln,gcd(d,n/d)=1

f(d)' g(n/d)

("unitary convolution") becomes a commutative algera with unit element E.

16) Dualize (4.7), which means: prove for x z 2 and any complex numbers wn the inequality

z

P-1

psx

Z IP X nsx.pin

W

-X

n

1. Z nsx

17) [DABOUSSi]. Prove

p`sx P k

.I

nsx,p°Iin

18) Assume that

w" -

n=1

x-1

w

12

n

I

nsx,p.i'n

s (4C +2)

x-1

nsx

1

Wn

12 S 2(C +1) 1

IW

n

X-1 I

nsx

I2.

n

1W12.

is absolutely convergent at the point

s = o1+it1. Prove that this DIRICHLET series is absolutely convergent

for every s = a + it If o Z a1. This result is not true, if the assumption of absolute convergence is weakened to convergence. In this case, prove convergence of Zn=1 in the region Re s > a1. 19) Prove Lemma 5.1. Hint:

Z- f(n) - IIpsx(1 + f(p) + f(p2) +

...

)

I

sZ n>x

If(n)I.

Tools from Number Theory

42

20) For integers k, in Re s > 1, prove n=1 cn(k)

'

n-s

2s-1 (k)

/

(ks-1

c(s) )

21) (SIERPINSKI, 1952). Let p1 < p2 < p3 < ... be the ordered sequence of all primes. Prove: 10-2" has a limit, say c. a) Z n=1 P. ' P. [102" c - 102 = b) The formula holds for n = 1, 2, ...

102 ' c

.

22) Define the polynomial p(x) by

p(X) =lsssn y 1 (x

e2

1

' n)

).

(s, n)=1

Prove

P(X)

=

I j ( xn/d - I

)µ(d)

din

23) Give the proof of EULER'S summation formula (Theorem 1.2)

In

detail.

24) Define D(f) by D(f): n H f(n) log n. Then the map D Is a derivation (so that D: C" '4 is linear, Ds = 0, and

D(f*g) = f*D(g) + D(f)*g).

25) g is completely additive if and only if the map f N f g is a derivation. Note that many properties of derivations are dealt with in T. APOSTOL 119761, §2.18. 26) Prove: For every positive integer k,

din dk = M + 1) nk' 2: r :1 c r(n)

and this series is absolutely convergent.

r (k+1)

Photographs of Mathematicians

43

PAUL ERDOS

J. KUBTLIUs

S. RAMANUJAN (1887-1920)

P. TURAN (1910-1976)

TURAN's photo, given to the first-named author by Prof. Dr. K. JACOBS, was already used in an article in "The Development of Mathematics from 1900 to 1950", Birkhauser Verlag (forthcoming 1994), edited by J. P. PIER. Birkhauser Verlag has kindly given permission to use this photograph again.

44

Photographs of Mathematicians

J. P. L. DIRICHLET

A. F. MOBIUS

(1805-1859)

(1790-1868)

G. H. HARDY (1877-1947)

J. E. LITTLEWOOD (1885-1977)

H. DAVENPORT (1907-1969)

4S

Chapter II Mean- Value Theorems and Multiplicative Functions, I

Abstract. This chapter mainly deals with estimates of sums over multiplicative functions and with asymptotic formulae for these sums. Rather simple, elementary methods lead to the mean-value theorems of WINTNER and AxER, in which multiplicativity does not play any role. Next, Inequalities for sums over prime powers are shown to be sufficient to obtain upper bounds for sums over non-negative multiplicative functions; lower bounds for such sums may be obtained under stronger assumptions. The HARDY-LITTLEWOOD-KARAMATA Tauberian Theorem Is em-

ployed to prove a useful theorem by E. WiRSING with some applications. Finally, following DABOUSSIs proofs, an elementary proof of the prime number theorem is given, and SAFFARI's result on direct decompositions

of the set of positive integers Is proved.

46

Mean-Value Theorems and Multiplicative Functions, I

II.1. MOTIVATION

Given an arithmetical function f: IN - C, the mean-value M(f) of the function f is defined to be the limit (1.1)

x-1

M(f) = lim

ao

X

Z f(n)

nsx

If this limit exists. In case f = 14 is the characteristic function of A, f(n) = 0 otherwise ), the some set .A4 of integers ( f(n) = if n 1

mean-value (1.1) is also the density S(4) of the set A: (1.2)

SW =

lim oo

X 1 E nsX IAW.

If f is a real-valued function, the upper Cresp. lower] mean-value M (f) Cresp. M_(f)] of f, defined as M (f) = lim Inf x-1 Z f(n), M (f) = lim sup x 1 X f(n), x--> o nsx nsx x- m always exists, and so the upper density S (A) and lower density 8_(4) of a subset A of N always exist. The density S(4) exists if and only If the upper and lower density of .A4 are equal.

More generally, often the asymptotic behaviour of the mean-value-function x H M(f,x) is required, where (1.3)

M(f,x) =

Z f(n), nsx

for example, if one is interested in results beyond the pure existence of M(f) = lim x _> m x 1 M(f,x), or if the mean-value (1.1) does not exist (this is the case, for example, for f = c, the divisor function). The existence of the limit (1.1) is often a disguised form of some other arithmetical statement, and thus It is of considerable Importance to obtain results on the existence (and the value) of the limit (1.1). For example: The assertion M(µ) = 0 is equivalent to the prime number theorem in the form fi(x) = x ( 1 + o(1) ), this being nothing more than M(A) = 1.

I,.1. Motivation

47

2)

n-2

Is a result on the density of the set of The result M(µ = 6squarefree integers. The result M(a) = ITZsksm (k), where a(n) denotes the number of non-isomorphic abelian groups of order n, provides some information on algebraic objects (much more precise information is available, also for other types of algebraic objects; see, for example, J. KNOPFMACHER [1975]).

Knowledge of Ensx t(n) provides information on the number of lattice-points (these are points In 1R2 with Integer coordinates) in

the planar region between the hyperbola -q s x and the axis. Denoting by r(n) the number of representations of the integer n as

a sum of two squares, then the behaviour of Ensx r(n) contains results on the number of lattice points in the disc B(O,x) [see 1.2, Theorem 2.7, and II.4].

If p(n) denotes the number of solutions of the congruence P(u) = 0 mod n, where P(X) is a monic, irreducible polynomial with integer coefficients, then the existence of M(p) is a non-trivial result concerning polynomial congruences.

FOURIER coefficients ?(a) =

of arithmetical functions and

RAMANUJAN[-FOURIER] coefficients ar(f)

=

are

defined via the notion of mean-value.

In the theory of sieve methods and their applications, estimates for sums such as 'nsx,gcd(n,k)=1 are useful. The question of the existence of a limit distribution (1.4)

lim N-1 s<{ n s N; g(n) N- m

sx

for a real-valued arithmetical function g is, according to the continuity theorem for characteristic functions, connected with the existence of the mean-value (1.5)

M(n H exp(

In the theory of uniform distribution modulo one, the foundations having been layed by HERMANN WEYL in 1916, the uniform distribution of a real-valued sequence (xn)n=1 ,2,... depends on the existence of the mean- values Mk = M( n H exp { 2ni k xn }) for every k In

Mean-Value Theorems and Multiplicative Functions, I

48

Z. The condition Mk = 0 for every k $ 0 is necessary and sufficient for the uniform distribution of (xn?n=t,2, .' Figure

11.1

shows the characteristic function of the squarefree 10 000. The small squares indicate the 100 (line at the bottom), 101, . ,200 (next line),

numbers In the range 1,

,

Integers 1, 9901,...,10.000 (top line)

,

,

It is easy to recmultiples of 4 and 25 in this diagram. Obviously,' most' ognize

of the integers are

squarlefreo.

fact, this is true, if 'most In

of means 'about 60 79 %' of all the integers 10

So

25

Fi gu re

II.1

7S

too

(Squarefree Numbers)

Numerous investigations dealt with the determination of mean-values for special arithmetical functions. The aim of this chapter, however, is, to provide general theorems which secure the existence of mean-values for large classes of arithmetical functions, or, at least, to provide estimates for the sum nsx f(n). Of course, it is easier to obtain results for classes of arithmetical functions which have some kind of arithmetical

structure, and thus the functions most frequently dealt with are multiplicative.

Multiplicative functions, according to I. (2.6'), are determined by their values on the prime-powers. Higher prime-powers are rather rare, their number up to x being (l.G)

zp. sx, kz2 l 5 L ps-/x l

+

3s kslog x/log 2

X3

(

X} ).

Therefore, it is reasonable to assume that the behaviour of >'n
II.2. Elementary Mean-Value Theorems (WINTNER, AxER)

49

is intimately connected with the behaviour of 2:PsX f(p). An important example here is E. WIRSING'S theorem, which is introduced in II, § 4. Further examples are provided by DELANGE's theorem and the theorems of P. D. T. A. ELLIOTT and H. DABOUSSI (see Chapter VI). The HALASZ Theorem (see section 5 of this chapter ) deals with complex-

valued multiplicative functions that are restricted in size ( the condition Ifi s I is assumed ), but there is no assumption on EPsX f(p). Multiplicativity is treated as a Tauberian condition. E. WIRSING published his similar theorems with other restrictions on the values of f just one year before G. HALASZ. In particular, his results (and, of course, the HALASZ Theorem, too) contain a proof of the famous ERDOS-WINTNER conjecture: Any multiplicative function f, assuming only the values 1, 0 , and

-1, possesses a mean-value. Choosing f = µ, the MoBIUS function, the truth of this conjecture includes the prime number theorem.

11.2. ELEMENTARY MEAN-VALUE THEOREMS (WINTNER, AXER)

Chapter 1, section 2, discussed, for a given completely multiplicative func-

tion h, the linear transformation Th :

C°V -_> CN

, Th : f H f * h,

of the vector-space CN = { f: N - C } of complex-valued arithmetical functions onto itself. The map Th is bijective, and its inverse is

Th 1: f H (w h) *

f.

Theorem 2.1. Assume that h is a fixed completely multiplicative arithmetical function with mean-value M(h) = H, and that the arithmetical function f: IN -) C has the property (2.1)

Z '=I n-1 ' I (Th-1(f))(n)

I

= Z '=I n-1 I

oo.

a

Then the mean-value M(f) exists and is equal to

50

Mean-Value Theorems and Multiplicative Functions, I

(2.2)

M(f) = H

X-=1 n-1

,

(Th-1(f))(n)

Choosing h = 1 (a constant function), we obtain the following result. Corollary 2.2 (A. WINTNER).

If

En 1 n-' -l

(2.3)

d

then the mean-value (2.4)

n

µ(d) f(n/d)I

M(f) = En=1 n-1'

din

exists.

Examples. (1)

Consider EULER S function 9 = µ * idN. Then din

d-1,V(d),

and so

(id)-1,9 = 1 s

(id-1-ii).

is convergent; hence Corol-

Obviously the series 2:n=1 lary 2.2 gives

n=1 n-2. IL(n) = (c(2))-1 = 6

n-2.

(2) If o(n) = Zd1n d denotes the sum of the divisors of n, then o = 1 * ld, and

(id)-1 * 1 . Arguing as before, Corollary 2.2 leads to M(

(id)-1,6

) = En=1 n-2 = I

'

n2.

for any d (3) If h is completely multiplicative , then h(n) = dividing n. Assume that M(h) = H exists and that Ih(n)I s K is bounded. Then, after a short calculation, Theorem 2.1, applied with f = h-9/id, gives the following mean-value result: H . Zn= n-1 d n H ' 21n=i

n 2'h(n)'µ(n) = H

n( 1- p-2.h(p) ).

(4) If h is the characteristic function of the set of integers coprime with and so some fixed integer m, then H = M(h) = n-1.p(n)

nsx, gcd(n,m)=1

=

`p(m)

m

, p.l' I (1 - p-2) m

II.?,. Elementary Mean-Value Theorems (WINTNER, AXER)

51

proof of Theorem 2.1. Abbreviate Th 1 ( f ) by f ' . Then f = h * f', and

nSx f(n) _ Ensx -

x

where O(d and

= o aX ),

dln

f'(d) h(n) rnl

dsx V(d)

nsx, n-0 mod d h` dl

asx f'(d)

( H.

X

as d

+

e(a\) ),

x - m; therefore IO(a I s s xa,

O (d) I s L(s) Is bounded, If d

if

x

d

2 K(s),

s K(s). Thus

E nsx f(n) _ 15d<m f (d) H- d + E1(x) + E2(x),

with two error terms; the first one is d>xd-1 f'(d) = o(x), as x --) oo,

E1(x)

due to the absolute convergence of d d sx

and so

E()I S 2

dsx/K(E)

I

f'(d)I -c-

dI+

/K(E)
The first sum is s d-1'If'(d)I, and is thus sufficiently small [using (2.1)]. The second sum is s L(s)

and so is <

/K(E)
a 5 L(E). x'

/K(c)
d-1'If'(d)I,

If x is sufficiently large.

11

If the function f In Corollary 2.2 Is multiplicative, then a nicer-looking result is available. 1) Corollary 2.3. Let f: IN -4 C be a multiplicative function satisfying (2.5)

z P

1)

f(P) - 1

< oo, I

This theorem will be sharpened later (DELANGE, ELLIOTT, WIRSING Theorems, see 11.4 and VII).

Mean-Value-Theorems and Multiplicative Functions, I

52

and (2.6)

2:

2:

P k22 P-k Then the mean-value

I

f( Pk )

I

< oo.

M(f) = rr ( 1 + P-1'(f(P)-1) +

(2.7)

P

exists.

Remark 1.

(a) If, in addition, f is strongly multiplicative (recall: this

means that f(pk) = f(p) for any p and any k = 1,2

M(f) = IT (

(2.7')

(b) If, in addition,

), then

+ P-1'(f(P)-1) ).

f is completely multiplicative, then

M(f) = Iii (

(2.7")

1

...

1+

P

Note that, in this case,

p-l.f(P))-1 ) p by condition

I

(2.6).

Proof of Corollary 2.3. The function f' = i*f is multiplicative, and so nsx

n-1.I f'(n)I s I-T Psx l 1 + P-1'If'(P)I + p 2 'If'(P )I

+

... ).

Using the values f'(p) = f(p) - 1 , and f'(pk) = f(pk) - f(pk-1), and utilizing the inequality I + (3 s exp((i) [if (i z 0], we obtain n-1.If'(n)I nsx

5 psx rr l 1+ s

+-7 p-k' I f(Pk) ka2

exp {psx I ( P-1.If(P)-II + I kz2

P-k.1

f(pk-1)I

)

f(Pk) - f(pk-1)I)} 11

The convergence of the series ZP follows from (2.5) and the convergence of p-2. Equations (2.5) and (2.6) imply the boundedness P of the series ( ZPsx( ... ) ) in the argument of the exponential function; thus the assumptions of Corollary 2.2 are verified, and Corollary 2.3 Is proven.

11

The method used for the proof of Theorem 2.1 also gives the following result.

Theorem 2.4. Let g, and h be arithmetical functions. Suppose that the mean-value M(g) exists and that the series Zn_1 is con-

0.2. Elementary Mean-Value Theorems

53

vergent. Then the function f = g * h has the mean-value M(f) _

(2.8)

n-1' h(n)

n

.

M(g).

The proof is left as an exercise (Exercise 2).

It is possible to weaken the assumptions in Corollary 2.2: absolute convergence of the series (2.3) is replaced by mere convergence; but an additional condition (2.10), that is weaker than absolute convergence is unavoidable. A sufficient condition is displayed in A. AxER's Theorem, as follows.

Theorem 2.5. Assume that f' = µ*f. Suppose that

f is

an arithmetical function, and put

the series Zn-1

(2.9)

Is convergent,

and that Y-nsN If'(n)I =

(2.10)

0(N) for N ->

ao.

Then the mean-value M(f) exists and equals

M(f) = Z'=I n-1'f'(n)

(2.4)

Examples. (5) The convolution formulae µ2 =

(

(1101). sq) * 1, id/cp

where sq is the characteristic function of the set of n-Z squares of integers, lead to M(µ 2) _ Z n=1 µ(n) = 6 n-2, and to x-'- Xnsx (n/p(n)) -3 IT (1 + (6) The function a r(n), where r(n) is the number of representations of n as a sum of two squares, is multiplicative and representable as = I

* (µz/gyp ),

{(p-1)p)-1).

Eden X(d), where X is the non-principal character modulo 4; therefore r = I * X, with convergent sum X d-1 X(d). So AXER's result gives the mean-value result I

M(ar)=1-1

t

1

1

3

5

..=a>t.

Remark 2. WINTNER's Theorem ( Corollary 2.2) follows from AxER's Theorem: The absolute convergence of X-=1 implies the following by partial summation (see I.1):

Z n-.If'(n)I nsx

.

n = x- nsx E

('x z n-1.lf'(n)I du s C'x. i

nsu

Mean-Value-Theorems and Multiplicative Functions, I

54

In fact, the same idea and sensitive handling of the sum and integral yields the stronger result Z W(n)I = o(x).

nsx

Remark 3. Condition (2.9) alone is not sufficient for the existence of M(f). See, for example, A. WINTNER [1943].

Remark 4. If M(f) exists and If 2:n n-1 f'(n) is convergent, then M(f) = Z n

Proof. The existence of M(f) implies M(f,x) = Z f(n) = nsx Partial summation gives

£(f,s) =

M(f)

n

o(x).

(s-1)-1, as s -4 1+.

Therefore, M(f) = lims-->

1(s) = limes

1+

1+

But the convergence of Implies, by the continuity theorem for DIRICHLET series (or, what amounts to the same, by partial summation), that lim9- 1+ E and Remark 4 is proved.

Proof of AXER's Theorem. Abbreviating (i - [3] by (a), a routine calculation (change of the order of summation) gives nsN

f(n)=E nsN Edin f'(d) N ' Ed-
dsN f'(d)

d-' f'(d) -

[3]

dsN f '(d)

'

l T}

Zd=1 d-1 V(d) + o(N) + R(N),

=N

with the remainder term (2.11)

R(N) = -d N f '(d) ' l N }

It must be shown that R(N) is not larger than o(N). The summatory function

MW, x) = Z f'(n) nsx

has the following properties:

11.2. Elementary Mean-Value Theorems

55

IM(f',x)I s 1 If'(n)I s

(I)

C by (2.10),

nsx

= o(x), as x -*

IM(f',x)I

o.

Statement (ii) is obtained from (2.9) and partial summation: (n-1.f'(n))

M(f',x) =nsx Z

.

n = nsx Z

y + o(1), (ii) is easily proved [in order

and, inserting Zmcx to show

du,

E 1 nsu

x

f x o(1) du = o ( ftx

1

x

du ), split the integral into f1

x +

Next we consider the sum R(M,N)

M
P(d)

{

n}.

ABEL summation II. 1, Theorem 1.3; Theorem 1.1, which is easier, is not

applicable, the function x H {x} being not differentiable] gives

N}

- R(M,N) = (M(f', N) - M(f', M<nsN-1

(M(f', n)-M(f',

n

Therefore,

IR(M,N)I 5 2

max

M<nsN-1

N

IM(f', n)I

M<nsN-1

I

Nn +T1 - [Nn- }I

Now divide the integers n in M < n s N-1 into two disjoint classes .' and

n

7: n is in P If [ N ] = C

n

otherwise it is in T. If n is in P, then

{jir}-{n}I-I

-n

Isn (n+1)

and so

M<nsN i,

nc ( M( f', n) - M(f',M)) s

max

M<nsN

IM(f', n)I

{

N n +T }

- {n}

I

n>M n n+1)

s N max M<nsN

`

M

N

For integers n ¢ T the inequality

1s n+f

I

n

IM(f', n)I.

Mean-Value Theorems and Multiplicative Functions, I

56

implies that there are at most M elements in J'; thus I{n-}-{n1I52

n 7

IR(N)I 5

s

IZdsM

f'(d)

a} I

6 NM max M<nsN

N

+I 2:M
and finally choose N so large (this Choose E > 0, then fix M = IM(f', n)I < E2 N. Then possible by (ii)) that max M<nsN

Is

11

11.3. ESTIMATES FOR SUMS OVER MULTIPLICATIVE FUNCTIONS (RANKIN'S TRICK)

This section deals with sums over non-negative multiplicative functions; the results given connect [estimates for] the size of values of f at prime-

powers in mean with the estimates of the sum Zn.,N f(n) from above (this Is rather easy) and from below (this is more difficult). The Ideas Involved are not too difficult: to obtain an upper estimate for EnsN f(n), where f z 0, an additional weight factor g(n) z I is introduced (for example, g(n) = log n If n z 3, or g(n) = (N/n)o if n s N), which makes the treatment of the sum easier (for example, log n can be split additively, or other factors g(n) make it possible to remove some troublesome conditions of summation without increasing the sum under consideration too much); the factor g(n) has to be chosen in such a way that the new sum EnSN f(n)-g(n) can be dealt with in a simpler way. Surprisingly that this simple method ("RANKIN's trick") is often very effective. Theorem 3.1. Suppose that for a non-negative, multiplicative arithmetical function f: N -> [0,co[, for every y z 1, the upper estimate

0 3. Estimates for Sums over Multiplicative Functions

P2]

S7

y (log Y),

f(pk) log pk s C

is true with some a 2 0 and some positive constant c1. Then there are constants c2, c3, which depend only on c1 so that for all x Z 2

Zx f(n) s c2 x (log x)'-' exp f p x P-1. f(p)

(3.2)

(3.3) Zx f(n) s c3 x log°`x exp { p21x p

+pZ k 2

P-k f(pk) },

E k2 P-k. f(pk)}. psx

Remark. If, for every prime-power pk, 0 s f(pk)s Y1 Y2k, where Y2 < 2, then (3.1) holds with a = 0.

Proof. The function n H log n is additive, and so log n = p"Iln log pk Inserting this into Y-nsx f(n) log n, inverting the order of summation, and using the multiplicativity of f, we obtain

I f(n) log n = p"sx E log pk E f(m) f(pk). nsx msx/P", P-l' M Inverting the order of summation again, and neglecting the condition this leads to the estimate Z f(n)-log n s 21 f(m) msx

nsx

p

"

7-

sx/m

f(pk) log pk s ci x (log x )°C

Z M-1-f(m). msx

Assumption (3.1) was used in the last line. The multiplicativity of f and the inequality I+ y s exp(y) imply

I m-1 f(m) s p11

msx

s exp

l (

+

I

P-1, f(p) + p

...

}

Z P-k' f(Pk)) = E(x), psx ki2 for the moment. Then, for x z 2, E f(n)-log(n) s ci x log°` x E(x), and

I

2snsx

Z psx

nsx

f(n) s s c1 x log°'-ix E(x)

n/ log 2) + { X-21

E

I<nsx

f(n)-log n/(2 log x)

log x/ log 2 + 2} S 4 c1 x logac-1 x E(x).

In our argument we used the fact that y H E(y) is monotonically increasing, and that the maximum of x- log x / log 2 in x 2 2 is attained at x = e2 and equals 1.06 ...

.

The estimate Zp5x p-1 s loglog(x) + c4 (see 1.7) implies the second assertion.

Mean-Value-Theorems and Multiplicative Functions, I

58

Another form of this theorem is given in G. HALL & G. TENENBAUM [19881, Theorem 0.1.

Theorem 3.2. Let f be a non-negative multiplicative function satisfying (3.4)

p s cIX ( X 2 1

P:CX E

),

and

log(pk) 5 c2.

Z P-k.f(pk) p k22 Then, for any x Z 1,

(3.5)

Z f(n)s ( c1 + c2 + 1) x

nsx

Z n-'-f(n). nsx

log-1 x

Remark. WIRSING'S condition f(pk) s YI'Y2k for k z 2, where 0 s Y2<2,

is stronger than condition (3.5) for the higher prime-powers. Proof. As before, we begin with nX

f(n)-log(n) = p4Z log (Pk)

s Z f(m) Z

psx/m

msx

.

ms

f(p) log p +

ZP`. P Y n f (M). f(Pk)

E plsx ,k22

f(pk) log pk Z

msx/p

f(m).

The obvious estimate

lmsx/p`

f (m)

SX'

P-k

.

m-1

msx/p.

.

f(m)

implies, together with the assumptions of Theorem 3.2, nZ

f(n)-log(n) 5 ( c1 + c2

x

Xx

m-I ' f(m),

and so, noting that ( log n + x/n ) z log x in I s n s x,

X nsx

f(n) s

I log X nsx

f(n) log n +

cl+c2+1)

x

log x

X log X ' mix

,

x

n f(n)

f(n) n

11

RANKIN'S trick is applied to the proof of the following theorem.

Theorem 3.3. Assume that f Is a non-negative multiplicative function,

and denote the maximal prime divisor of n by P(n) = pmax(n). Suppose that z > y, and that for some A > 0 the series

0.3. Estimates for Sums over Multiplicative Functions

p Yk22

(3.6)

S9

pk. f(pk). pkA

is convergent. If

log( log z/logy / log y --> 0

(3.7)

as y -- oo, then

log z [ E P-'-f(p) -log( llog z_)]). log y p5y

(3.8) n>z,P(n)sy n-'-f(n) << exp( log Y

Remark. The series n,P(n)sy

n-1. f(n)

is equal to II (

1+

P-1. f(p) +

... ),

P:CY

and so, in some cases, Theorem 3.3 also implies results on -1

1nsz,P(n)sy n

.f(n)'

Proof of Theorem 3.3. With some parameter 8, 0 s S s A, we use a weight-function, printed in bold-face, to obtain n-1 .f(n) 5 En>z,P(n)sy (n/z)8

C+n>z,P(n),y

S z-sz

n-1 .f(n)

n-i,f(n),ns n P(n)sy

z-s,TI (I + psy

p-2,f(p2),p2s +

<< exp {-s.log(z) + Epsy A good choice for 8 is 8 = ( log y )-

log z

log ( logy ). Owing to condition (3.7) this expression is < A if y Is sufficiently large. Using this choice of 8, the assertion of Theorem 3.3 is proved. 0

Lower bounds for non-negative multiplicative functions are accessible only with greater effort. First we deduce an auxiliary result on "rather small" non-negative multiplicative functions, where, for simplicity, the summation is extended only over squarefree integers. We need some elemen-

tary results on prime numbers with explicit constants. The inequality

n p < 4n

(3.9)

(valid for n =

psn 1,

2, ...) is easily proved by induction (see Exercise 17).

DENIS HANSON, Canad. Math. Bull. 1S, 33-37 (1972), gives a stronger

Mean-Value-Theorems and Multiplicative Functions, I

60

result). Equation (3.9) implies 0(n) < 2-log 2

n.

Partial summation leads to

Z p-flog p < 2-log 2

psx

s 1.4

(1-log 2)

log x + 2-log 2

log x + o.45

< 1.55

log x,

if x z e3. In passing we mention that, according to RossER-ScHOEN(and therefore x 2 e3 FELD [1962], psx P-1-log p < log x for x > could be replaced by x Z e). 1

Figure 11.2 shows the rather smooth function (log x)-1.psx Z in the range I s x s 600.

p

1

20

200

100

400

300

Soo

600

F i g u r e II.2. Lemma 3.4. Let h be a multiplicative, non-negative arithmetical function satisfying 0 s h(p) s 0.1 for every prime p.

Then, for x Z e3, (3.10)

Z

nsx

µ2(n)

'

n-'- h(n) Z 0.8

fl psx (

1

+

P-1. h(p) )

.

Proof. Denote by P(n) the maximal prime divisor of n, as before. Consider the difference between the sum under consideration and the ex-

11.3. Estimates for Sums over Multiplicative Functions

pected approximation 1T

IT (1 + p-'-h(p)) Then,

applying

(

1 + p-I h(p)

µ2(n)

RANKIN's

n-1-h(n) =

.

n>x P(n)Sx

n 1 h(n)

s (log s (log s 0.1

x)-I -

P x

n1

the

h(n) Z log p / log x Pin

p-1 - h(p) log p ' P(a)S X

x)-1 Z p1. h(p) . log p- II ( Psx Psx

(log x)-,

n-1,h(n).

log n log x

,

n>x,P(n)sx E µ2(n) n>x,P(n)sx

µ2(n)

weight-function neglecting the condition

with

again,

idea

log n /log x, this difference is (writing n = d i x/p and using h(p) s 0.1 µ2(n)

61

(

1.SS

log x)

1-1 (

1

2(d) d-I h(d) 1

+

p-1. h(p)

+ P-1 h(p)

if x 2 e3, and the assertion will follow, after replacing 1.55 by 2. Theorem 3.5 [BARBAN]. Let g be a non-negative multiplicative arithmetical function bounded at the primes, 0 s g(p) s C1.

Then there Is some positive constant C2, depending only on C 11 such that the Inequality (3.11)

n

Z

r

µ2(n).n-1.g(n)

2 C2

exp ( N P-1'g(P) P

holds as soon as N is sufficiently large.

Proof. In order to apply Lemma 3.4, choose an integer m so large that m = [10-C I I + 1, and put z = N1/r". Define completely multiplicative functions g* and H0 by (3.12)

g*(Pk) = {g(p)}k, k = 1, 2, ...,

H0(n) =

m-0(n)

.

n-1 .g*(n)

If H is any non-negative, completely multiplicative function, then

Mean-Value-Theorems and Multiplicative Functions, 1

62

H(n) }m = Zz ,..., n . H( n s rszm Z H(r) t m (r)

with the divisor function tm(r) counting the number of representations

of r as a product of m factors. The values of tm at primes p are tm(p) = m, and thus tm(n) = m0(n) if n is squarefree. Using the representation tm = 1 * ... * 1 = 1 * 'cm-11 the relation (3.13)

tm(Pk)

k+m -1 =l m- 1 ),k=1,2,...,m=1,2,...,

is easily proved by induction (Exercise 16).

Write r in the form r = r'-d, where d is squarefree, r' is 2-full (this means that pir' implies p2Ir' ), and gcd(r',d) = 1. Then, neglecting the condition gcd(r',d) = 1, we obtain

Ensz H(n) }m s Xr' szm, r' 2-full H(r) tm(r') psz^'

1

2: dSzm

H(P3)'tm(p3) + ...

+

Ed:,.,,

x

}

i

With the choice H = HO given above (see (3.12) ), and paying attention to g* (p) s C1, the product PO = 1f 1 + HO(p2).tm(P2) + HO(p3).tm(p3) +

...

}

is convergent (for this, an estimate such as tm(pk) «m'e pk is useful). The sum G(N,m) = { nsN-

j2(n).n-1,g*(n),m-n(n) }rn

={E

O (n)

nsNvm

satisfies

E µ2(d) H0(n) }m s P0 d sN G(N,m) 5{nsN"m E

.

d-1, g(d)

on the one hand; on the other hand we obtain, from Lemma 3.4, G(N,m) 2

( 0.8 )m , n pSNvml

These two estimates Imply the relation

1

+

g(p) )m.

}=n

19.3. Estimates for Sums over Multiplicative Functions

dsN

µ2(d)

d

(0.8

g(d) z P o

1

)m P

n (I +

63

\n,

p

m

l

for every N 2 e3m. For 0 s x < 1 the inequality z (1-x2

(1+x) = (1-x

is valid. Thus we obtain (with x = g(p)/mp)

TI (1 - (

P1

)2 ) > IT

(

P

P

1 - 0.01. p-2 1J > 0.9

(by an easy numerical estimate), and, finally, d

µ2(d)

N

d

z po

-I.p1m.

( 0.8 )n'

.

exp(-C1

x exp ( psN E

N

psN p

1

m

The well-known asymptotic formula for Zpsx P-1 (1.6, (6.7)) yields N ,gym
P-1

= log(m) + o(1), N -- w,

and thus Theorem 3.5 is proved. BARBAN's Theorem dealt with functions of order n -I ..in mean". For multiplicative functions of order 1 "in mean", removing the restriction of summation over squarefree integers, one gets the following theorem. Theorem 3.6. Let f be a non-negative multiplicative arithmetical function, satisfying (3.14)

f(p) z Y1 > 0 for all primes p z POI

and

f(pk) s Y2 for every prime p and every k = 1,2,...

Then, with some positive constant y depending only on PO ,

(3.15)

the inequality Z Z nsx f(n) 2 Y ' x ' exp ( psx

holds for every x > x0. Proof. As in the proof of Theorem 3.1 we begin with

1))

.

Y1' Y2

and

Mean-Value-Theorems and Multiplicative Functions, I

64

nsx f(n) log n =

L. (x)

2

msx f(m) 2:

p sx/m, p m

f(m) I psx/m, Pkm f(P)'log p.

msx

Assumption (3.14) on the values f(p), and a TCHEBYCHEFF-estimate for S(x), yield

L (x) i f

msx, m squarefree

f(m)

{ 2

Y1

Xm +

0(1) - YZ6)(m) log X},

the constant hidden in the O-notation depends only on Po, Y1 and y Since ca(m) log x s t(m) log x = O (m' log x) = o (x/m) for m s xi, we obtain Lf (x) 2 Y 3 '

ms-x

f(m) . xm. µ2(m),

and BARBAN's Theorem 3.S gives

Emsrx µ2(m)

.

m-'-f(m) 2 .y3

,

exp{

Zps&

p-1 = loglog x + Y4 + 0(1), and

,rx
p-1 f(P) 5 Y2' log 2 + o(1)

(for these results, the prime number theorem is needed) we obtain finally (for x 2 xo) nsx f (n) 2 (log z

x)-1 ,

Lf(x) Z YS

log x exp{zpsx P-1' f(P) }

, y , x , eXp { Epsx p-1 (f (P) - 1) }.

This is the result we looked for.

0

11.4. Wlrsing's Mean-Value Theorem

6S

11.4. WIRSING'S MEAN-VALUE THEOREM FOR SUMS OVER NON-NEGATIVE MULTIPLICATIVE FUNCTIONS

A rather general, and very useful, result concerning the behaviour of sums En-<x f(n) over non-negative multiplicative arithmetical functions is a E. WIRSING's Mean-Value Theorem [196t]. The main assumption is one specifying the asymptotic behaviour of the sum Zpsx f(p). The HARDY-LITTLE WOOD-KARAMATA Tauberian Theorem allows deduction of a result on 2:n ,x n-1 f(n) (in fact, in his paper [1961] WIRSING used elementary arguments, but in [1969] he sketched the method used here in [19691). Elementary arguments establish a connection between the sum 2:n-1.

f(n) log n and Insx n-

1

f(n).

Theorem 4.1 [E. WIRSING]. Let f: N - [0,co[ be a non-negative multiplicative function. Assume that with some constant t > 0 the asymptotic relation (4.1)

t + 0(1))

p

psx

x, x --> co

holds, and that for every prime p and k = 2, at prime powers are 'small', f(pk)

(4.2)

3, ...,

the values of f

sY1Y2 k, where 0< '2<2.

Then, as x -> oo, the asymptotic formula

(4.3) nsx Z f(n) _ (1 + o(1))

e-et

x

log x

I

-,

I' (t)

T1 psx l

+ f(p) + f(p2) p

p2

+

...

holds. i° denotes EULER's constant, r(.) the gamma-function. Remark 1. (4.2) may be replaced by weaker assumptions, e.g.

of (log x)-1 ),

E P Zk=2, P"2x and

f(p) =

O(pl-g

) for some 8 > 0.

Using the Relationship Theorem from Chapter III (Theorem 2.1), these assumptions can be weakened further.

Mean-Value-Theorems and Multiplicative Functions, I

66

Remark 2. (i) Starting with (4.1), partial summation gives (4.4)

log x,

P-1'f(p)'log p = ( t + o(1))

psx

and, furthermore, the convergence of the series (4.S) (Flog P)-1 f(p) = f I(t + o(1)) 2 + log u du .

P

u

2

p = O(p), and this estimate, together with (4.S),

(ii) (4.1) gives

implies the convergence of

Z P-2' f 2(p) << Z (p . log p)-1 . f(p). P P

Lemma 4.2. Suppose that the assumptions of WIRSING's Theorem 4.1 are valid. Abbreviate (4.6)

nsx

f(n)

and

by M(f,x),

-7 n-1 f(n) nsx

by m(f,x).

Then, as x -* CO, (4.7)

t + 0(1))

M(f,x)

Proof. Put Z(f,x) =

nsx f(n)

'

log x

m(f,x).

log n. Then, making use of the multiplica-

tivity of f (as in the last section, 11.3, but with a little bit more care): 8(f,x) =

nsx f(n)

2:

'

P"Iln

pk

log Pk = Em,P"S X, Pt m pk -

Em,p"sx

pk

Z f(m)'{p"sx/rn I

Z

M:rx

p""sx/m

p.i'm,r=1,k21

Using assumption (4.1), the first sum t

.

x M

+

Z

f(pk).log(pk

p"sx/m

)

equals

0( m x ) + p"sxTm,k=2

Thus the problem is reduced to an estimation of remainder terms. Having proved the formulae (4.8')

:=

S 1

p"sx,kk2

o(x)

11,4. Wirsing's Mean-Value Theorem

S

(4.8")

:_

2

67

o(x),

Z

p.., s x

rxl,k21

we obtain

E(f,x) = E

msx

t'

lm

f(m)+Om

= t x' m(f,x) +

msx

0

x m

= t x m(f,x) + o(x m(f,x)). Partial summation (see I. Theorem 1.1) then gives the assertion [the Integral f 2 t m(f,u) log-2u du is easily shown to be of lower order, using the trivial estimate m(f,u) s m(f,x)].

In order to substantiate (4.9) one has to show that, for some [fixed, large] constant K, the estimate E m-1 f(m) = of x m(f,x)) x/K<msx

X,

holds. But this sum is [the bold-face factor in the next formula is 2 1, as long as x/K s m ] 5K

Zx/K<msx f(m) s K' 'msx f(m)

.

log(m) /log(X/K)

= 0K( 2(f,X)/ log X) = 0K(

log x).

For S1, the sum up to x)-2 is easily estimated by C')(x/log x), using the weaker condition of Remark 1. The remaining sum, where k 2 2 and

x is less than

f(p) log x P = of

2-2 x)

x l

log( 1 gx -1 = o(x).

k

The sum S

2

:_

E p...sx r21,k21

Is more difficult. Split this sum into S' + 2S" + S'", with the following summation conditions:

S': k = r = 1, p2 s x,

Mean-Value-Theorems and Multiplicative Functions, I

68

S":r=1,k22,1+k5x, S"':kZ2,rZ2,pr+k5x. Since f(p) log p = o(p), S' _

Ps/z o(P)' f(p) = C7( x'(log

x)-1

).

For S", fix an integer K so large that (1+K)-1 + (log Y2/log 2) < 1; this is possible by the condition Y2 < 2. Then split S" into S"1 + S"2, where 2 5 k 5 K in S"1, and k > K in S"2, and use f(pk) 5 YIY2k. Then S"2 5 upsxv,,. , f(p) log p <<

-Y

P:""Wc,

f(p)

'

Y1, Ikslogx/log2 k- Y2

k

log2 (X). xlogy/log2 < < x1-3

with some S > 0. More easily, we obtain S"I << ZP_.CX- f(p) log P' Z2sksKk 12k << xi-S

The treatment of S". with the result S"' << x1-s is simpler than that of S" and is left as an exercise (see Exercise 12). By Lemma 4.2 it is sufficient to prove an asymptotic formula for m(f,x) = nsx n-'-f(n). The following will be proved, applying the HARDYLITTLEWOOD-KARAMATA Tauberian Theorem.

Theorem 4.3 (E. WIRSING). Denote by g: W -> [0,0o[ a non-negative mu!tiplicative function satisfying (4.10)

GPsx

t + o(i)) log x, where t > 0,

g(p) log(p)

g2(p)

(4.11) P

and, for all primes p and k = 2, 3, g(pk)

(4.12)

...

,

5 Yl'(Y2/P)k, where Y2 < 2.

Then

(4.13)

n 21

g(n) _ (1 + 0(1))

I'(t+1) r -L°t

_ (1 + 0(1))

1'(t+1

IT (1 + g(p) + g(p2) + ... psx '

P

exp (

psx g(P)),

)

11.4. Wirsing's Mean-Value Theorem

69

with a convergent product

P = II exp -g(p))'

(4.14)

(1

+

g(p)

+

g(p2)

+

... )

.

P

Remark 3. Condition (4.12) may be replaced by the weaker assumption

E E g(pk) < oo. p ka2

(4.12')

proof of Theorem 4.3. Consider the generating DIRICHLET series

Vg,o) _ Yn=l g(n)

n-O.

The following conditions hold: (i) (ii)

(iii)

The product P is convergent. £(g,o) is convergent in o > 0. o-t B(g,o) ti P exp ( H(o)

),

o -* 0+,

where

H(o) = Y_ g(P)'P ' - t

log(o-1).

(iv)

The function x H L(x) : = exp ( H(1/x)) is slowly oscillating.

(v)

Z g(p) P

(vi)

For any real r, 0 < r <

P-a =

'

g(p) -

X

psexp(1/o)

o(1), as o -4 0+.

1,

Z g(p) = rlog(C ) + o(1), y ->Co.

yr
Taking these assertions for granted, (iii) and (iv), with g z 0, permit application of the HARDY- LITTLE WO OD- KARAMATA Tauberian Theorem (see

Appendix A.4) to the DIRICHLET series £(g,o); thus we obtain [using (v) in the 2nd line]

nI g(n) - r(P 1) ti

(log x )t .

P

r(t+1)

'

e

et

exp { P g(p) p-1/Iog(x) - t loglog x}

exp {PSx L g(p) }.

Assertions (i) to (vi) remain to be proven.

O) First abbreviate exp( -g(p)) (1 + g(p) + g(p2) + ... ) to w(p), and choose p0 so large that g(p) s z for p 2 p0 (this is possible; g(p) tends to zero). For 0 s E s 1 [for example, by TAYLOR'S formula] the inequality

Mean-Value-Theorems and Multiplicative Functions, I

70

ESexp(-e) s1-E+2 E2

is valid; therefore, with some 0 = 0p w(P) - 1

I

= I(

5

1 - g(P) +

,

0 s 0p

s

1,

(1 + g(p) + g(p2) + ...) - 1

2

I -g2(P) + (1-g(P)).(g(P2)+...) + 11.3 shows the functions 1-x,

Figure

exp(-x), and 1-x+2'x2

for 0 s x s 2. The Figure

diagram was produced using the computer algebra system

11.3.

RIBMANN II.

The convergence of the series g2(p) and g(pk) and the inequalp k22 P ity Ig(p)I s 2 [for large p] imply the absolute convergence of the product

P = n (1 + (w(P)-1)) (ii) Because of the inequality (here multiplicativity is exploited ! ) g(n)-n-a

Z nsx

5 psx n (1 +

g(P)'P-o

+ g(p2),p-2o + ... )

for o > 0, deduced by partial the convergence of the series summation from (4.7)!) and of 2: p2: k22g(pk) imply the convergence of . (g,o) in a > 0. So (ii) is true. (vi)

is proved by partial summation: {log

y,
log(y) - log(yr)

=

p}-1

{log y }-1

+ t . f y { log u - log yr + o (log u) } {u log2u}-ldu, r and, performing an easy calculation, this equals

-t

log r +

o(1).

Next we come to (ill), (iv) and (v). As long as a >

0,

jj,4.Wirsing's Mean-Value Theorem

log £(g,(j)

71

p = 7,1 (a) + 7,2(x), Z g(P)'P o + E h (a)

p

P

where g(p2).p-2o + ... )

hp (a) = {log li +

-

I.

x Z 0, y Z 0. Figure 11.4 gives the function

TAYLOR'S formula gives I log(1+x+y) - X1 s y+

2(x+y)2 if

log(1+x+y) - x 0.0 0.0 0.4 0.1

y + ''(x+y)2 in the region x,y > 0.01. It was produced using the com-

puter algebra system RIEMANN II (Begemann & Niemeyer, Detmold).

2 F i g u r e

11.4.

Therefore,

I

hp(a)

I

S kk2 g(pk) + 2 (k 1 g(pk) )2.

By the WEIERSTRASS criterion, 7,2(6) is uniformly convergent in 6 Z 0, and so 7,2(0) = 7, 2(0) + o(1), a - 0+.

7,1(6) is to be treated by partial summation. Split this sum into 7,1(x) _ Ipsexp(1/o)

g(P)'P-o

7,11(6)

+ 'p>exp(1/o)

+

7,12(6).

The second sum is easier to handle: 7,12(6) =

Je /6 { 1/aE

g

By assumption (4.7) the expression in braces {

...

} is

t

(log u - log exp (6-1)) + o(log u), and thus, using the substitution w = 6 log u, a straightforward calculation results in 7, 12 ( 6 )

t + 00) )

fa` 1

w-1

dw.

Mean-Value Theorems and Multiplicative Functions, I

72

For 211(0), we apply partial summation to the difference: P-°

(0) 11

psexp(1/°) g(P) log p

psexp(1/0 g(p) _

1/0

_

f e PsU I

pse 1/0

log p d P'u( lo°g u)du.

2

p, and sub-

Using the asymptotic formula (4.7) again for Epsx we obtain stituting w = 211(6) - Zpsexp(1/°) g(p) = O(1) -

[

f 1 w-1,(1-e-W) dw. O

Summarizing, using the integral representation 1

dw - f

L° = f

0

OD

dw,

1

for EULER'S constant, we obtain the formula g(P)'P_°

= 2:psexp(1/0) g(p)

Zpsexp(1/o)

+ 0(1)'

and (v) is proved. The function x H L(x) = exp ( H(x-1)) is slowly oscillating: without loss

of generality, c s 1; then, using the definition of H(.), (v) and (vi),

log L(x) = H((cx)-1) - H(x-1)

log

exp(x)
= clog C -

g(p) - clog x + 0(1) c + 0(1) = 0(1).

So, finally, as a -) 0+, collating our results, .D(g,O) = exp ( X1(6) + X2(0) )

=P where H(O) =

0-L P

exp (

P

0

exp ( X1(0) -

H(O) ),

1

is slowly oscillating. The proof of

Theorem 4.3 and 4.1 is now complete.

0

We note that E. WIRSING also proved results on complex-valued func-

tions and quote his result without proof.

11.4, Wlrsing's Mean-Value Theorem

73

Theorem 4.4. Assume that f satisfies the conditions of Theorem 4.1 with t * 0; let f : IN --j C be a second multiplicative function, If s f, I

satisfying

I

(4,15)

p ti

psx

S

x, x

m.

and

IT(

(4.16)

1

psx

+

P_I.f*(p) + =

of psx IT (1

+ P-1'f(P) +

Then

X

(4.17)

f*(n) = o (

nsx

f(n) ).

nsx

Remark 4. Condition (4.16), concerning the product TI from the simpler condition

(...), follows

f(p) - Re{f*(P)}) = oo.

(4.16*) P

There are numerous possible Applications of WIRSING's Theorem, some of which are presented below.

(1) Denote by b the characteristic function of the set (3 of positive integers, representable as a sum of two squares: B _ { n e N; 3 x,y e 7 L : n = x2 + y2 }, 6(n) = 1, if n e 8, b(n) = 0 otherwise.

Then 6 is multiplicative by virtue of the Identity (x2 +

-I2) = (x +

Obviously, b(2k) = 1, b(p2k) =

1

y)2.

for any prime p and any k e N. From

elementary number theory it is known that b(p) = 0, and 6(p2k+1) = 0 If p = 3 mod 4, and 6(pk) = I for any k e N if p = 1 mod 4. Thus, Z psx

p = 0(1) +

psx, p=1 mod 4

log p = zx + o(x),

by the prime number theorem (see 1.7 (7.12) ). Now WIRSING's Theorem 4.1 is applicable and yields

Mean-Value Theorems and Multiplicative Functions, I

74

x

1

Z 6(n) = nsx

r(z)

.I(l+p-i+p-2+...) 2rT 1+p-2+p-4+...1,

log xp=1(4) psx

p=3(4)

Note that F(2) = it. The product over the primes p = 3 mod 4 is convergent, and thus x (1+p 1) rT nsx 6(n) - Y log x psx,p-1 mod 4 Y

x

exp

x

E psx,pn1 mo

p

)

Y

Figure II.S shows the function x-i -Tnsx 6(n) (upper curve) and the approximation, given by the right-hand side of the formula at the top of this page, in the range 1 s x s 9oo. Only every third value is plotted.

O.5 .

0.1 30

300

150

600

F I g u r e

900

H.S.

(2) Given a monic, irreducible polynomial P(X) e 7L[X] of degree z denote by p(n) the number of solutions of the congruence

1,

P(m)=0mod n, Ismsn. Then p 2 0, p is multiplicative, 0 s p(p) s deg(P), p(pk) is bounded by a result of T. NAGEL(L) ([1919], [1923], see [1956], p.90) and the prime ideal theorem implies psx

P(P) ' log p - x

(see ERDOS, [1952]). Thus, again, WIRSING's theorem is applicable, giving

z P(n) ^'

nsx

e-

x

log x

(l + P-1 . P(P )+ rT P:9 X

p-2 .P(p2) + ...).

11 4.

Wlrsing's Mean-Value Theorem

7S

k) (3) The divisor-function t is multiplicative, t(p = k+l, and for any real

rz0

Z tr(p) log p - 2r

x.

psx Thus,

WIRSING'S theorem gives a much sharper result than the upper

estimate deduced in Chapter I (Theorem 2.7 (c)): tr(n) ^, C2,6 (F(2r)-t , 1g (1+ p -t.2 r+ P-2. 3 r + ...)

znsx

Y'

1gx

t\

exp (2: psx

x)2'

(4) Given some fixed set 7 of primes with the property log p ti -c-x, t > 0, we denote by No the set of positive integers solely composed from primes in 7,

neA1o#* (pin=* p.5r), and the characteristic function of this set by f7: n -->1, if n e Al o, 0 other-

wise. Then there is an asymptotic formula for Ensx f, (n), and, for any fixed integer m, the values c)(n), where n e AZo, are relatively uniformly distributed modulo m; this means that the number M r m s.(x) of integers n s x, n in Y, with the property w(n) = r mod m, is asymptotically equal to m-t tt{ n s x, ne A 0 Denote

by fU (n)

an

s

f (n) n) = t = and

£c,,

()

m-r.u). Put then the conditions of Theorem 4.3 are fulfilled,

m-th root of unity

(abbreviated

Yp p- -( fSr(p) - Re( f*(p) ) ) = w, if E $ 1. WIRSING's Theorem 4.1 gives an asymptotic formula for MS.(x) =nE f (n),

and Theorem 4.4 (which Is not proved in this Chapter) gives

E f (n)

nsx

s`°(n)

= o ( Mf(x) ), if e $ 1.

But

M

a,,r,m,J (x) _ nsx

f(n)

m-t

Eo(n)-r

,

a m-r.u

Mean-Value Theorems and Multiplicative Functions, 1

76

=

m-1

Mf(x) + 0 ( Mf(x) ),

and so the result is proved.

0

U.S. THE THEOREM OF G. HALASZ ON MEAN-VALUES OF COMPLEX-VALUED MULTIPLICATIVE FUNCTIONS

E. WIRSING's Theorem 4.1 is not sufficiently strong to give a solution for the old conjecture of ERDbs and WINTNER: any multiplicative function taking only the values +1, -1 and 0 possesses a mean-value. But WIRSING believed that an assumption such as p ti

psx

'C

-

log x

instead of (4.1) ought to be sufficient for deducing [some] results on the mean-value of a multiplicative, real-valued function f. One of his results, 1967, is quoted below. One year later G. HALAsz dealt with complex- valued multiplicative functions of modulus less than or equal to one. However, for complex-valued functions difficulties may occur. The example f(n) = exp(

n ),

with

Z f(n)

nsx

(1+it

)-1

x

xit + 0( log x )

shows that for t $ 0 the mean-value of a complex-valued function of modulus 1 need not exist (see Exercise IX, 1). E. WIRSING [1967] proved the following results. Theorem 5.1. Let f be a non-negative multiplicative arithmetical function, satisfying

If(p)I s G for all primes p,

(5.1)

(5.2)

psx

P-1

f(p)

log p - t log x,

The Theorem of G. Halasz

u .s.

77

with some constants G > 0, t > 0, and

2 If(Pk) I < ao; p k22 P-k ' then, in addition, the condition

(5.3)

If 0 < t s

1,

E

(5.4)

p kk2, p`sx

If(Pk)I = C7 ( x/log x).

is assumed to hold. Then (5.5)

21

nsx

f(n) _ (1 + o(1))'

e

log x

-°t

r(-0

. IT

+

f(p

psx l

f(p2)

+

+

p

P

...

.

Theorem 5.2. Assume that f: N -4 C is multiplicative and satisfies assumptions (5.1) to (S.4) of Theorem 5.1, and, moreover, that Z

(5.6)

p-I

.

(

If(P)I - Re (f(p))

)

< co.

P

Then assertion (5.5) of Theorem 5.1 is true.

Some preparations are helpful for the formulation of the next result. Let

g={ z= p'e1" E C; O s 9< 2n, O s p s r(p) } be some region in the complex plane C containing 0. Define its "average radius" by

r(g) _

(2n)-I

.

f2n r(y) dp. 0

Theorem 5.3. Assume that f: N -) [0,co[ Is multiplicative and satisfies the assumptions of Theorem 5.1. Let f : N -4 C be multiplicative, If s f. Suppose there Is a convex region g c C with average radius r(g) < 1, containing 0, such that contains all values f*(p). Then I

fz

nsx

(n) =

x

log x

e->°t

r(t)

rr / psx l

1

+

(p) P

+

f (p?) p2

+

...

+o

nsx

We do not prove these theorems here, but refer to WIRSING'9 paper [1967]; a proof by A. HILDEBRAND for a special version of WIRSING'S

Theorem for real-valued functions is given in Chapter IX, and, in the

Mean-Value Theorems and Multiplicative Functions, I

78

same chapter, we give a result due to G. HALASZ1) [1968] for complexvalued multiplicative functions of modulus IfI s 1; the proof given there will be "elementary" and follows H. DABOUSSI and K.-H. INDLEKOFER [19901.

Theorem S.4 (G. HALAsZ). Let f be a multiplicative arithmetical func-

tion of modulus IfI s 1. Then there exist a real constant a, a com-

plex constant C and a slowly oscillating, continuous function L: [1,co [ -) C, ILI = 1, for which the asymptotic relation

Z f(n) = C

nsx

L( log x )

xl+Ioe

.

+ o(x)

Is true.

The function L and the constants a, C may be given explicitly. 2) proof of parts of Theorem S.4 is postponed until Chapter IX.

The

11.6. THE THEOREM OF DABOUSSI AND DELANGE ON THE FOURIER-COEFFICIENTS OF MULTIPLICATIVE FUNCTIONS

In 1974 H. DABOUSSI and H. DELANGE announced the result that, for irrational values of a, FouRIER-coefficients f^(a) of multiplicative arithmetical functions f of absolute value IfI s are zero. DABOUSSI and DELANGE [1982] proved the following stronger result. 1

Theorem 6.1. Let f be a multiplicative arithmetical function for which the semi-norm (6.1)

1)

2)

IIfII 2 2

: = lim sup x --> co

x-1

' nsx E If(n)I2

In the authors' opinion, GABOR HALASZ's method, a skilful variant of the method of complex Integration, seems to be definitely simpler than WIRSING's method of dealing with convolution Integrals. See also the paper by K.-H. INDLEKOFER [1981a].

II.S The Theorem of G. Hala sz

79

Is finite. Then, for every irrational a, the mean-value (FouPJERcoefficien t)

f (a) = M(

(6.2)

lim

x - eo

°C

x-1

E f(n) e2"Ian nsx

is zero. We do not give DABOUSSI and DELANGE's proof, but sketch a proof of a result which is a little weaker - the relationship theorem of Chapter III allows the deduction of Theorem 6.1. The result is as follows.

Theorem 6.2. Denote by TA the set of multiplicative functions with the properties If(p)I s A for all primes p,

(6.3' )

(6.3")

n

N If(n)12 s

for all Integers N Z 1.

Abbreviate by Sf(c() the exponential-sum

S (a) _ z. f

nsN

Then S() = o(N), as N - co, If f e S'A and a Is irrational. Remark. Based on the "Large Sieve", H. L. MONTGOMERY and R. C. VAUGHAN [1977] prove a stronger result:

If f Is in SrA, then, for q s N and gcd(a,q) = 1, (6.4)

Sf(a/q) <
2N)-1

+ N (9(q))-' + (qN) ( log(2N/q) )1+1

uniformly for all functions f in Sr A; the implied O-constant depends only on A. They deduce the

Corollary 6.3. Let I a- g I s q-2, where gcd(a,q) = 1. If 2 s R s q s N/R, then

Sf(a) << A N (log

2N)-1

+ N (log R)1+' R I.

Proof of Theorem 6.2. The "large sieve inequality" [ see Appendix] gives

IqSQ I.

(a,q)=1 IZ M+1snsM+N an

ein _q l2 lI

Mean-Value Theorems and Multiplicative Functions, I

80

(N+Q2

<<

)

M+1snsM+N Ian

.

I2,

for complex sequences a n . This implies

PQ

p

p

n sN

f(n) e (an )-

=P Q p I p-'-2: f(n)e(an) - M:9 EN p n _< N =p'Q Z p-

e(b/p)

P-2

Z

e(b/p)

(am

) I2.

lsbsp

21

m N

Using the CAUCHY-SCHWARZ inequality for S 2: pSQ p-1 2:

lsbspl

(am )

ms N ,m-= O ( p )

...

I

I,2

this

2

is

(am )12,

Z1sbsp I >msN

and the sieve inequality gives the estimate

«(N+Q2) rn'N 1

If(m)I2<<

A

N

2

If

QsNil,

Summarizing, we obtain p

I

PSZQ

P

1,N-1.Sf(a) - N-1 _YmsN,m=-O (p)

if Q s N. Next, replacing with an error of O(p PSQ

p

,

LmsN,meO(p)

(am)12< <

1

f(m) by lmSN/p

we obtain, as long as Q s IN, (amp) IZ << 1.

N-'- ZmN/P

I

The CAUCHY-SCHWARZ inequality now implies f pQ

I

where P(Q)

f(a) - N-'- ZsN/ M p

p Y_

psQ

(amp) I

p-1. Abbreviating

D(a) s f I ZpSQ ...

I

}z <<

P(Q),

by D(a), and writing

} + I 2:PSQ N

1.

msN/p ...

I,

we obtain P(Q) )'

+

N-1

I XpSQ

ZmsN/p

I.

The square of the last double-sum over p, m is dealt with by the CAUCHY-SCHWARZ inequality:

11 7.

Application of the Daboussi-Delange Theorem

{ 1 2:psQ

(amp)

msN/p

81

}2

I

(amp)}2

[ ZmsN IpsQ,pSN/m s

GmsN I ZpsQ,pSN/m f(p) e((xmp) N

The

pSQ

I2

= N Em Zp...{Ep,...}

1msmin(N/p,N/p') e( am (p-P') )

p,SQ f(p) (p')

contribution of the diagonal elements (p = p') to this sum is sN.

2: pSQ

If(p)12

. N/p << N2

P(Q).

The non-diagonal elements p * p' contribute << N

Q2

where, obtained by summing the geometric series, m = min IN, 1maxQ {

I

sin na'h

I

}-

Summarizing, for Q s N3, with some constant y, depending on A only, D((x) 5 Y

{ P(Q) }-'

{

1

+ (M Q2 )'

}.

Given a large K, choose Q with the property P(Q) Z K2.Y2. Then s I If N is large; the irrationality of a is exploited here. Our estimates then give D(a) < K-'-2.

11.7. APPLICATION OF THE DABOUSSI - DELANGE THEOREM

TO A PROBLEM OF UNIFORM DISTRIBUTION

A sequence {xn} of real numbers is termed uniformly distributed modulo 1, abbreviated u.d.mod 1, if for any pair a, (3 of real numbers, 0 s a < 0 s 1, 11mN

N-1 .

u { n s N; a s (x n) s (i } = a - a,

Mean-Value Theorems and Multiplicative Functions, I

82

where {x} = x - 1x] denotes the fractional part of x. According to H.

WEYL s criterion, a sequence {xn} of real numbers is u.d.mod 1, if and only if N-1 lImN -4. EfSN exp ( 2nik-xn) = 0

for every integer k $ 0. Theorem 7.1. If g Is a real-valued additive function, and If a is an irrational real number, then the sequence xn = g(n) -

is uniformly distributed modulo 1. Proof. Given an integer k * 0, the multiplicative (!) function f: n H exp{2nik

g(n)}

fulfils the assumptions of Theorem 6.1 (or 6.2). Therefore, the meanvalue M( f e_koe) = 0. Written in full, limx

_4

x 1 - Ens. exp ( 2nik

g(n) -

0,

and so, according to WEYL, the sequence g(n) - an is u.d. mod 1.

a

11.8. THE THEOREM OF SAFFARI AND DABOUSSI, I

Let the semi-group IN of positive integers be represented as a direct product of two subsets A and 2, IN=A xB, so that every n is uniquely representable as a product

n = a- b, aE4,bE 2. In terms of the characteristic functions a of .4, p of 2 (a(n) = n c A, a(n) = 0 otherwise), this property is equivalent to

1

If

II 8. The Theorem of Saffari and Daboussi

1

83

= a * p.

(8.1)

The question as to whether a "direct factor" A of IN possesses a density was answered in the affirmative by B. SAFFARI [1976a,b] and P. ERDOs, $. SAFFARI and R. C. VAUGHAN [1979]. SAFFARI proved the following theorem.

Theorem 8.1. Assume that IN = A x B Is a direct product, and

E b-1 <

(8.2)

bc$

ao.

Then A has a density S(4)

b-t 1-t

Y_

bc,8

J

Of course, the density of 3 is zero. The case where Y-be2 b-t = oo was treated in the [1979] paper quoted above. In this case the density S(A) also exists and is zero. We give DABOUSSI's proof [1979] of Theorem 8.1.

Given y z 2, define completely multiplicative functions ZY and sy, the characteristic functions of integers composed of Earge, resp. small, primes, by 1 ,

(8.3)

ZY(p)

Then Z *s Y

Y

0

= 1, Zn

+

if p> y if p 5 y

sY(p) -

n-1. s (n) = IT ( PsY

Y

1

1,

if p 5 y

0,

if

+ p-t + p-2 +

>

P

... )

y

< oo, and the

mean-value

M(Z) = M(l * s -t( ) = M(1 *(its ) ) _ (7- n Y Y Y >

(n) )-t

1.s Y

exists. This is proved using WINTNER'S Corollary 2.3, for example.

Define aY = (s

Y

Y

Y

(n/d). Then the mean-value

M(( ) = (EbcR b-t,s(b) )-t

(8.4)

co, this follows from Theoexists for every y 2 2. Since Z rem 2.4 for example, and, by multiplicativity, the result n-I.sY(n) (

)-t

. C n-t

,

(a'sY)(n)

is easily transformed into (8.4). Next we show that

Mean-Value Theorems and Multiplicative Functions, l '

84

M(a,x)

(8.5)

Gnsx a(n) s M(oeY,x), for x z

:=

1.

Taking this for granted, the theorem is proved as follows: a*a = plies lbc.8, bsx M(a, x/b _ Ensx 1 = [x], and so M(a,x) +

bc.X3, 1
1 inn_

_ [xl.

M(a, x/b

Therefore, making use of (8.5),

[xl - Ebc2, 1
(8.6)

b-1

In the case where 2: bc,$

)

s

M(a,x) s Woe ,x).

= co, by (8.4), we obtain

lim sup x-1 M(a,x) s lim sup x m x -* _ and, letting y -- co, S(.) = 0.

Y

,x) = ( 2

be,$

Y

(b))-1,

If Zbc2 0 < co, then (8.6) gives

1-( bcR,b>1 i

b-1)

lim

x-m

x -3 m

Y

s lim sup

x

x

Using limx y -4 oo, we obtain

x M(a,x)

,x)

Y

(b) )-1, and then letting

(E b-')-' s lim inf x1 M(a,x) s lim sup x -1 M(a,x) s (Z x -3 oo

x --> oo

be.

b-1)-1

be.8

It remains to prove lnsx a(n) 5 Z.!x aY(n) (this

is (8.5)). Since, by

complete multiplicativity, the relation sY (a (3) = sy, a * sY p holds, we s obtain s Y

X a(n) = nsx

a(n)

(

s a*s a * Z Y

nsx

Y

)(n)

Y

Eabc&x,acA,bc.s s (a)

y

csx/a

$

y (c) 21bsx/ac,bcR sy

5 lasx,ac.r1 sY(a) Ccsx/a Zy(c) = Here

1nsx(sY'a*tY)(n)

= M(ay,x).

1 1.9. Daboussi's Elementary Proof of the Prime Number Theorem

s

Zbsx/ac,bc2

8S

s 1,

used. This is true, since, given a number of the form ac, there is A e s4 [otherwise, if b'-ac = A' a s4, at most one be. for which A' b, which is impossible, since IN is a direct prob' $ b, then was

duct of .vl and B ].

11.9. DABOUSSI'S ELEMENTARY PROOF OF THE PRIME NUMBER THEOREM

The Ideas used In 11.8, worked out in greater detail, make It possible to give an elementary proof of the prime number theorem (DABOUSSI [1984]).

Theorem 9.1. Put M(x) = Y-nsx µ(n). Then lim

x -- m

x-1

M(x) = 0.

Proof. With some [large] parameter y we use the completely multiplicative functions 8y, sy defined In 11.8, where ty(p) = I for "large" primes p > y and sy(p) = 1 for "small" primes p s y. For brevity, write M (x) = nsx µ(n) s (n). Then µ = µE *µsy, and so y

y

y

M(x) = 7nsx

y

(n) My (x/n).

the finite sequence of squarefree integers composed only of prime-factors p s y; in x/d +1 < n s x/d the I function My (x) is constant. Therefore, with d q+1 = oo, x/dq+1 = 0, we n Denote by d1 = 1 < d2 <

...


9

obtain, using (9.1), (9.2)

M(x) = E1515q My(dI)-y_x/d,.,<nsx/d, ey(n)-µ(n).

The mean-value M(Zy) = W 1-p 1) exists. Dividing (9.2) by x, we obtain

Mean-Value Theorems and Multiplicative Functions, I

86

S psy II

lim sup x

ao

E IMy(d i

(1-p-1),

1/d

1SJaq

1

1+1)

(9.3)

IT pay

(1-p-1) f

-2.1M (t) I y

1

dt,

again using the remark preceding (9.2). Put y)-1'

C = lim (log

IT

(1-p-1)-1

pay

y-+ 00

and a = lim sup x-1-IM(x)I. Then 0 s a s 1. We are going to prove the x -moo

following.

(i) There exists a constant 8 Z 1 such that for any (i In a <

< 2 the

Inequality

f1 t

(9.41)

dt s 0.8-1

-

log y + 0(i)

Is true. If a > 0, then 8 > 1. f°° (9.411)

y

t

dt s

log y + o(log y).

1)

Having proved the Inequalities (9.4i), (9.411), and making use of inequality (9.3), we obtain as

and so, for p - a+, a = 0. Therefore, the prime number theorem is proved and so it remains to prove (9.41) and (9.411).

for any a, I s a < b, the integral

Proof of (9.41). It is known, that a

is bounded. [The value 6 is of no importance; any fixed bound M is sufficient. See Exercise 18.] If M(x) does not change sign in 11, y], then fly

y

(01 dt = f Y1

dt s 6,

and we are ready. In the other case, we prove (9.41) using (9.5)

1>

8 = min (

2,

1

+ a2/(24) ).

In fact, It Is easy to improve the remainder term In (9.411) to 0(1).

11.9. Daboussi's Elementary Proof of the Prime Number Theorem

87

Given (3, a < P < 2, then IM(x)I s 13x, if x z xp. It suffices to show

f b t2

(9 6)

IM(t)I dt s (P/S)

log(b/a)

for two consecutive zeros a, b of M(x) in xp s a < b s y. According to the size of b/a we distinguish three cases: lot case: log(b/a) z 6

(S/P). Then, trivially,

f ab t 2

IM(t)I dt s 6 s (P/S)

log (b/a).

2nd case: log(b/a) < 6 (S/P) and (b/a) s (1-0)-1. Since M(a) =0, for every t in [a,b]

IM(t)I = IM(t) - M(a)I s It-al s 2

(9.7)

(3

t.

Again, this estimate implies

f 8b t 2 IM(t)I dt s 2

fab

t l dt s 2P

log(b/a) s (0/8)

log(b/a),

which is (9.6) in the second case.

3rd case: log(b/a) < 6 S/(i and b/a > (1-20)-1. We apply inequality (9.7) in the interval a s t s a and we apply in [xp s ] the estimate IM(t)I s s t s b. This leads to

fab t 2

IM(t)I dt s 2P

log(1-2P)-1

+ P log((b/a) (1-2P))

= P log(b/a) + 0 log(1-2P).

Using the definition of S and 2P log(,-2, 3) 5 - a P2 5 - : a2 we see that

2P log(1-2P) 5 - 6 (8-1) 5 - (1-5-1)

.

P

log(b/a),

and so we obtain (9.6) in the third case also. Proof of (9.4ii). The following auxiliary functions are needed: (9.8)

F(x) = fox U-1

(1-e-u) du, where x > 0,

Mean-Value Theorems and Multiplicative Functions, l

88

k(s) = fo e-ax , eF(x) dx, where s > 0.

(9.9)

It Is obvious that the function s '- k(s) Is positive, decreasing and continuously differentiable. Furthermore, fss

s k(s) -

(9.10)

1

k(u) du = 1 for any s > 0.

[The left-hand side of (9.10) equals - f 0co eF(x)-sx

dx f Oco -A-

(F'(x) - s) dx

e

F(x)-sx

dx = 1.

I

For fixed y 2 2 we consider the function h, defined in x > 1: 1O

y)-1

1

kl

h(x) _ (log

(9.11)

Then, for any x > 1, (9.10) leads to the equation log x

(9.12)

h(x) = 1

Jxyx +

u-1

h(u) du.

Partial Integration gives (9.13)

5"

x-1

(

fxyx

u-1

h(u) du ) dx = f z (2-u)

k(u) du

log y.

Lemma 9.2. Denote by C the limit given in the displayed formula Immediately following (9.3). Then

k(u) du = C -

f12 (2-u)

1.

Proof. Starting with the convolution relations log = A* 1 , ands

y

and using the abbreviation S y(x) = nE

Sy(x)

sy(n)

log = ( sy A) * s y ' n sx

sy (n), we obtain

log n = d sy(d) A(d) :cx

'

Sy l

d

)

log x = -x sy(n) log n +n-7 sy(n) log(x/n)

2

PSy.P sx

yp

nsx y

log X. n

II.9 Daboussi's Elementary Proof of the Prime Number Theorem

89

Therefore,

lo-Px h(x) dx = 27

(9.14) fy SY(x)

log p SY(

P)

hXX) dx + R1 + R2,

with the remainder terms Rl =

(9.14.1)

f-y E

/ xk) h(x) dx, x2

log p

P&Y

Syl p

k> 2, p`S x

R2 = fY

(9.14.2)

log(n) hXX) dx.

sY(n)

The error terms R1 and R2 are bounded: the estimate ' J1

u-2.S(u)dus nsx I n-1s(n)_ IT PSY Y

(1-p-1)-1=O(logy)

Y

Implies

Rl s h(y)

p-k log p , f °° u-2

7-

P, k>2

1

SY(u) du = d(1),

and

R2 = C'0 ( h(y)

I n_1 n-1 sY(n)) = O(1).

Starting with the elementary relation nSY p-1 log p = log y + 0(1) [see I. (6.6)], partial summation yields the formulae (9.15.1)

(9.15.2)

p-1 log p

P E

Y/t
h(pt) =

p-1 log p

ftt

U-1 h(u) du + 0(h(y)), for t z y,

h(pt) = fYt u-1 h(u) du + 0(h(y)), for t > 1. Y

The Integral on the right-hand-side of (9.14) equals Z log p f m... dx, and using the substitution x H p

fy//P

t-2 h(pt)

SY(t)

p-1 dt

for this integral. Interchanging summation and integration, we arrive at

fYm SY(x)

12JIyx

Y h(x) dx = fY t 2 S (t) ft

+

f' t -2.S y

(t) Y

Y/t
p-1-log

E p-1-log Wry

Using (9.15.1) and (9.15.2), the right-hand-side changes Into

0(1).

Mean-Value Theorems and Multiplicative Functions,

90

fl t-2.Sy(t) . ( yfYt

I

(ftYt u-1 h(u) du) dt +

O(1).

(9.16) fy Sy(x) . log2x dx = f y x 2 S(x) (f yX u-1 h(u) du) dx +

C'0(1),

h(u) du) dt +fy t

Using (9.12), we finally obtain y

The left-hand-side of (9.16) becomes floc x-2

Sy(x) dx -fly x-2 Sy() dx = .

(9.17.1)

n-1

s y (n)

-

p-11-1

n=1 (1

= P:CY

- 2y

I

-

n-1 sy(n) + d(1)

2:y

n-1 + O(1)

_ (c + o(1) - 1) logy + 0(1). Using S (x) = x + 0(1) for all x s y, and (9.13), the right-hand-side of y (9.16) becomes

fy x-1 ( fy " u-1 h(u) du ) dx + f y 0(x-2 f y' u-1 h(u) du ) dx + 0(1)

= f2 (2-u) k(u)

(9.17.2)

du

log y + O(1).

Dividing (9.17.1) and (9.17.2) by log y, then as y --) co, Lemma 9.2 is proved.

The proof of (9.4ii) can now be concluded, using very similar ideas. Starting with the convolution relations A * µ = - µ log, and (sy A ) * (sy (1) = - sy µ log, we obtain IMy(x)I

log x s

sy(n) A(n)

IMy(x/n)I + 21 s (n) log(x/n), nsx Y

and so

-

fy IMy(x)I log x x-2 h(x) dx = f log P. IM (x/p)I x-2 h(x) dx + 0(1), y pSx Y and

fy IMy(x)I

x-2 dx = fi x-2 IMy(x)I

. (fyx Y

u 1 h(u) du

dx + 0(1).

Using My(x) = M(x) for x s y, and estimating IM(x)I by 3x, if x 2 we finally obtain

xP,

1I 10, M. Nair s Elementary Method in Prime Number Theory

f

IMy(x)I

x-2dx

91

u 1 h(u) du ) dx + 0(1) = a J('1 x-1 ( ( ('YX ly

Y

=a

ft2 (2-u) k(u) du logy + 0(1) = p (C-1) logy + 0(1),

and so (9.411) is proved.

11

11.10. MOHAN NAIR' S ELEMENTARY METHOD IN PRIME NUMBER THEORY

The prime number theorem implies the asymptotic formula 41(x) 'Z x2, asx --boo,

(10.1)

where 41(x) = %sX fi(n). On the other hand, by elementary arguments, (10.1) implies the prime number theorem 4(x) - x. M. NAIR [19821 gave a simple method for obtaining good lower estimates for 41(x). Theorem 10.1.

If x is sufficiently large, then 1(x) z 0.47459

(10.2)

x2.

...

By a more elaborate calculation, NAIR was able to improve the constant 0.47459 to 0.49517, which is rather near the best-possible constant We do not deal with this improvement here. Lemma 10.2 (CAUCHY). The determinant D n of the n-rowed matrix (ai l )1,J=1,...,n with elements aw = (ai + bl)

where ai, bl are given real, positive numbers, has the value Dn =

l n (ai - al) (bt - b) tsi 1

i

(ai +

b)- 1

Mean-Value Theorems and Multiplicative Functions, I

92

The Proof is given In POLYA- SZEGO, "Aufgaben and Lehrsatze aus der Analysis'; II, 7. Abschnitt, Aufgabe 3 (p.299). It is by inductive arguments 0 (see Exercise 19 ). Lemma 10.2 is used with a1 = m + i, bl = j for some positive integer m. Then D

n

=

IT (i-j)2( rT(m+i+j))-1

1sJ
1,J=1

A good approximation for Dn may be obtained with the aid of STIRLING'S formula

log Dn = - 1 (2n+m)2 . log(2n+m) +

(m+n)2.

log(m+n)

(10.3)

n-z

+

m + Co ((n+m) log (n+m))

(see exercise 20). Denote by do the least common multiple of 1, 2, ..., n: d n = lcm [

1,

2, ...

,

n 1.

The connection with prime number theory is due to the relation 4(n) = log dn.

Multiplying Dn by the product IT 1s15n do+m+1, we obtain a determinant with integer entries and a positive value. Thus

log D. + z1sjsn which may be written as

4(m+n+1) Z 0,

1(m+2n) - cy1(m+n) z - log D.

In the special case where m = obtains for large n the relation

with some constant s > 0, one z GI(s)

n2 + C7( n log n),

with the function -1(s) = z (2+s)2 log(2+s) - (1+s)2 log (l+s) + 2 s2 log s.

Therefore,

41(x) - 1((1+s)x

(2+s)-1)

z

x2 + C)(x log x),

1I 10. M. Nair's Elementary Method in Prime Number Theory

93

and splitting the interval [l,x] into O(log x) intervals of the shape p], we obtain by summation 41(x) z i'(s)

(2s+3)-I

,

x2

+ C)( x logzx ).

The choice s -> 0+ leads to the lower bound (note that

2 log 2

3

0.462)

41(x) z 0.46

x2, if x is large.

Some numerical calculations (see the graphical illustration in Figure 11,8) show that the maximum of the function ii(s) (2s+3)-1 occurs near s = 0.22, with the "good" value ui(s) (2s+3)-1 = 0.47459 ... . Figure

11.8

was produced

using

the program system RIEMANN II.

Figure 11.8

11.11. EXERCISES

1) Let h be a completely multiplicative arithmetical function with meanvalue M(h), which is bounded at the primes, Ih(p)I s y for every

prime p. If f is a multiplicative function with the property that the two series E p-1 p

f(p) - h(p)

I

< oo, and 21p 2:k22 p-k

f(pk)

I

<m

are convergent, then the mean-value M(f) exists and is equal to

Mean-Value Theorems and Multiplicative Functions, I

94

M(f) = M(h) pn (

1+

k21

P-k

( f(pk) - h(p) f(pk-1))).

Hint: use Theorem 2.1. 2)

Prove Theorem 2.4 in detail. Hint: x-'- -7nsx 17(n) = _ n=1 n-1. h(n) - M(g) - En,x n-t h(n) . M(g) n-1 -

nsx

h(n) o(x/n).

3) Let h be a bounded, completely multiplicative function with meanvalue M(h). Then Mid-1

-h

M(h) Zn=1 n-2 h(n) = M(h) IT

(1

P

- p-2. h(P))-1

4) Define the sets 7 and e by n E IN, n squarefree },

Sr

e

(m,n) E IN2, gcd(m,n) = 1

}.

Prove that the mean-value M(17) exists and equals

x-2 m,nsx XZ

lim x 5)

1

L' (m,n).

Prove that the density of the set

A = { n E IN: the number of primes with p2In is even

exists and is equal to a rl ( 1 -

}

2-

P

Hint: the function 2

1A - 1 is multiplicative. Use Corollary 2.3.

6) Prove that for any non-negative integer t the density of the set

Dt={nEIN,(j(n)sZ} Is zero.

7) Prove that the frequency with which the digit m c

{

1,

2, ..., 9

}

occurs as first number in the sequence 2n (written in decimal scale), is logto ( 1 + m-1 ). 8) For every direct product IN = ,r7 x 2, both the series

Z a-1 acA

b-1

bcS

cannot be convergent Is it possible that both series are divergent?

95

11.11. Exercises

9) Let a, Ko a N. Does there exist a direct product IN = A x .V, such that (a)

ai`, k = 0,

A

1,

2, ...

}?

(b) A{ak, 0s k 0, such that for every real number a > 0 and every integer N > 1 there exists a multiplicative function f, where Jfl s 1, satisfying

(logN)-1.

l2:fSNf(n) 11)

a) Define

g(n) _ : e { (x,Y) e 7L2, x2 + y2 = n }, g' = µ * g.

It is known that g' is the non-principal character modulo 4. Use this to prove M(g) = ;

it.

Prove that f is multiplicative, and has mean-value

(b) Let f =

M(f) =

3 27t

i - 2 (p(p+1))-1 ). IT Pal (4) (

12) Prove the estimate for S"' given in section (II.4).

13) Assume that f is a multiplicative arithmetical function, satisfying f(p) z c1 p 3/2+E for every p rime p, with some positive constant cl. Prove that En,. µ2(n)

(f(n))-1 << x 1

Hint: apply RANKIN'S trick.

14) Let a be real and N Z 2 a positive integer. Define a completely multiplicative function f by : f(p) = c If p s 2 N, f(p) = e(-(xp) if

N < p s N, f(p) = I if p > N. Prove that there exists a complex constant c Icl = 1, such that f is in TI and has the property ,

log-1

lSf(a) l >> N

N [see Theorem 6.2].

IS) Prove:

a)

cp-1

'

b) nsx c)

id = (cP-1 . µ2) * 1. (p(n)-1

n )2 = C) (x ). nsx n cp-2(n) = C'1( log x ).

Mean-Value Theorems and Multiplicative Functions, I

96

16) Prove the formula tm(pk)

k+m_

m -1

1

k, m = 1,

2, ...

.

17) Prove by mathematical induction the inequality n p < 4N. psN

l is odd, use n p s ( 2kk+ 1 psN

Hint: if N = 18) In 1 s a < b, prove

)

rI

P :r k+1

p.

t2 Gnstµ(n)dtl 56.

Hint: use I. Cor. 2.5.

19 ) For positive real numbers ai, bi, where 1 s

i, j

s n, calculate the

determinant

Dn = det(a + b

1s1TI n

(a, - a l

) (bt - bj) ( IT (ai + b

Y

Hint: subtract the n-th row from another one, and extract suitable factors. Then subtract the n-th column from another one. Proceed by induction on n. 20) Prove formula (10.3), using STIRLING'S formula

log(n!) = n -log n - n + O(n).

97

I

0

Chapter III

Related Arithmetical Functions Abstract. The simple fact that multiplicative functions are determined by their values at the primes leads to the Idea that multiplicative functions which do not differ "too much" at the primes behave similarly. The aim of this chapter Is to render these vague Idea more precise and to provide a universally applicable result In order to reduce proofs to the simplest possible assumptions. The notion of "relatedness" Is a measure for "not differing too much at the primes". Our result states that two related functions f and g, which are not too large, are con-

nected by a convolution formula g = f * h, where the function h Is small In the sense that the series Z lh(n)l ' n-1 is convergent. This chapter is close to the paper HEPPNER & ScxwARZ 119831.

Related Arithmetical Functions

98

III.1. INTRODUCTION, MOTIVATION

Multiplicative functions are determined by their values at the primepowers pk for the relation f(

IT pk) =

f(k)

J_ Pk1In

PkOn

Higher prime-powers pk, where k 2 2, are rare: the number of these

up to xis 1 = C7(xg,),

pksx, kk2

and so one is inclined to conjecture that multiplicative arithmetical functions behaving similarly at primes have similar properties". This chapter aims to give an exact meaning to these vague formulations. The theorem we are going to prove will be Important for simplifying proofs by reduction of these to special cases which are easier to handle (for example, multiplicative functions may be replaced by com-

1)

For example, given two multiplicative arithmetical functions f and g, which behave similarly at the primes, one might ask for conditions that ensure one or more of the following assertions: If f has a mean-value M(f) = Urn x-1 E f(n), then g has a meanx->

nsx

value, too.

If f has Fourier-coefficients f ('y) = M(

then the Fourler-coeffl-

cients of g exist. If the RAMANLUAN-Fourler-coefficients ar(f)

do

_

exist, the same is also true for the function g. If the RAMANLUAN-expansion 2.1sr<m ar(f) cr(n) of f is Cpolntwise, absolutely, uniformly, ... ] convergent, the same is true for g. If the series E1sn<, n- 1.f(n) is convergent [or summable by some .

method of summation], then the same result Is valid for

lsn<m

If the function f is In the class .Yl' of functions which are arbitrarily near with respect to the semi-norm IIfII = lim supx-, to finite linear combinations of RAMANLUAN sums, then g is in 2'.

.

111.1. Introduction, Motivation

99

pletely multiplicative functions).

The set Al of multiplicative arithmetical functions (not identically zero, and so f(1) = 1 ) is a commutative group (Al,*) under convolution

f*g: n H

7-

din

f(d)

g(n ),

'

we have seen in 1.2. The unit element is given by the function e, with values e(1) = I and E(n) = 0, If n > 1; the convolution-inverse of f is denoted by f-1(*), and it is possible to determine the values of as

f-I(*)

recursively from the relation f * f-I(*) = S.

The "near-ness" of multiplicative arithmetical functions at the primes Is measured using the following definition. Definition. Two functions f, g in AZ are called related, if the sum 21 P-I P

f (p) - g(p)

I

< CO

Is absolutely convergent.

The sum 7- p-t is divergent. Therefore, for related functions f, g, the P differences f(p) - g(p) have to be be rather small, at least in the mean, to ensure that f and g are related. In spite of the rarity of the higher prime-powers we need some restrictions on the size of the functions under consideration. We define the subset ; of AZ as the set of I

I

multiplicative arithmetical functions satisfying

If(p)I2<00: E Z P_ k. If(Pk)I
(1.2)

P

p

2 If(p)I2 < co and E E P-k' If(pk)I < oo}. p k22

The following notation will frequently be used in this chapter: the set (1.3)

;*

=

{f a ;, Y (p,s) t 0 in Re s 2 I for every prime p} f

is the set of those multiplicative functions in ; for which cpf(p,s) does not vanish in the complex half-plane Re s z 1, where W (p,s) denotes f the p-th factor In the EULER product for the generating DIRICHLET series £(f,s) = n2I f(n)-n-' of the arithmetical function f, and so

Related Arithmetical Functions

100

(1.4)

f

(p,s) = (

I +

The set of multiplicative arithmetical functions f with absolutely convergent series E.,,1 is denoted by

Afro = { f

U.S)

e

Ii

n-1

;

n21

I f(n) I

< oo }

(absolutely convergent, multiplicative); and, finally, we put (1.6)

Re

f e0; f

related toe } _ { f e §; E p-1 P

If(p) I

< w I.

Examples.

(1) If f is multiplicative, then the functions f and µ2 f are related. (2) The functions n H n-1, more generally n H n-a, where a > 0, are in Re and, at the same time, in ,v4L°l11.

(3) The functions n y n 1 p(n), n H µ2(n) and n H µ(n) are in Every bounded multiplicative function is in ;. (4) If f is a multiplicative function in ;, then the multiplicative functions fcm and fsm, defined by f(p)k for every prime p and every k 2 1,

fsm(pk) = f(p) for every prime p and every k 2 1, are completely multiplicative (resp. strongly multiplicative) are in and are related to f if If(p)I < p in the first case. (5) Another important construction ("multiplicative truncation") is the following: given a multiplicative function f In jy°, then, with a fixed squarefree integer K, f may be written as a convolution product

f = f(K) 1

where f(K) and

*

f(K) 2

f(K) are multiplicative, and 2

f(K)(pk) = f(pk), if pIK, and = 0 otherwise, and f(K) (pk) = 0, if pIK, and = f(pk) otherwise.

The function f21q Is in ; again, and it is related to f; the function fiK)

is in Re as well as in AIM. An important special case of this construction is the choice K = 2 3 5 ... P with some fixed prime P.

111.2. Main Results

101

111.2. MAIN RESULTS

Using the notation introduced in III.1, the main result of this chapter is the following theorem. Theorem 2.1. Assume that f and g are multiplicative arithmetical functions

which are related. If f is In §* and If g is in §, then the multiplicative function h, defined by g = f * h, has the properties (a) (b)

E n-1

n21

Z n-'

Ih(n) I< oo, so

h E 946Al.

h(n) = II p (p,s)

na1

p

g

)-i

( Pf(p,s)

In Resk1.

Remark. The formulation of Theorem 2.1 is not symmetrical in f and g;

of course, h = g * f-1(*) is multiplicative, but in order to be able to show the absolute convergence of I n 1 h(n), one must derive some additional properties of f: the non-vanishing of the factors of the EULER product implies nice properties of the convolution inverse f -l(*).

Example. In the formulation of Theorem 2.1, the condition f E

cannot

be replaced by f c ;°. The function g = µ (Modus function) is and the multiplicative function f, defined by f(pk)

-2, if p = 2, and k =

1,

if p > 2, and k =

1,

-1 , 0 ,

in

,

if k z 2,

is related to g, and is in but not in An easy computation shows that f-1(*)(pk) = 1, if p $ 2 and k is arbitrary. For p = 2, however, we obtain f-1(*)(2k) = 2k; therefore

h(pk) = f-1(W)/p ) - f-1(*)(pk-1)

=

2k-1

and so I n-1

Ih(n)I

if p = 2,

is divergent.

However, in spite of this example, a condition weaker than f sometimes sufficient for applications.

E

§°*

is

Related Arithmetical Functions

102

Theorem 2.2. Assume that f and g are related, and that both are in Assume, furthermore, that the factors Epf(p,s) of the EuLER product of the DIRICHLET series D(f,s) are not zero In the closed half-plane Re s Z 1 for every prime p outside some finite exceptional set h°r,

Then there exists a multiplicative function h in .AL°Ai (see (1.s)) satisfying g = f * h, provided f(pk) = g(pk) for every exceptional prime p e 9? and every k = 1,2,

...

.

m

Theorems 2.1 and 2.2 are deduced from the following theorem. Theorem 2.3. (1)

The set § Is closed with respect to convolution: f*g Is in ;, if f and g both are both In

Figure

III.1

.

(2) The set §* is closed with respect to convolution.

(3) If f is In §°*, then the convolution Inverse f-'(*) Is in (4) The set AeAi of functions f with absolutely convergent series n 1'If(n)I and the set RE of functions In §, related to the unit element s, are Identical. An extension of Theorem 2.1 to functions that are related in some (apparently) more general sense is easily possible.

Given p, 0 < R s 1, the multiplicative functions f and g are called 0-related, if ZP P-13-1 f(p) - g(p) is convergent. In analogy with notation given earlier we use the abbreviations I

(2.1)

.A 'J1ip =

(2.2)

Re0 =

{

{

f f

E

Ai; X n a If(n)I < co

E ;p; f a-related to E

{f E At, Z p- 213 If(p)i2 < m and E

(2.3)

},

p kl. If(pk)I <

and (2.4)

b°13

={f

Theorem 2.4. (1) The set

cpf(p,s) $ 0 In Re s Z Q for every prime pl. Is closed with respect to convolution.

(2) The setp is closed with respect to convolution.

111.2. Main Results

103

(3) If f is In ;a*, then f-1(*) Is in ;a*. (4) The sets s4G'At a and RE0 are Identical.

Corollary 2.5. If f c ;°p and g c ;p are 13-related, then the multiplicative

function h = g *

f-1(N)

is In AL°At p.

Theorem 2.4 and its corollary are reduced to Theorem 2.1 and 2.3 by applying these results to the functions f": n H n(1-a).f(n), g": n H n(1-O),g(n). An important generalization is due to L. LuCHT (preprint [1991]). The

use of weights enables him to deal, for example, with more general remainder terms. , by Theorem 2.3 (3) (this theorem will be proved In the next section) the inverse f-1(*) is in , and so, using Theorem 2.3 (1),

proof of Theorem 2.1. Since f is in

h=

f-'(*)

* gc

If p is prime, then h(p) = g(p) - f(p); f and g are related, and therefore Y-

P

p-1.Ih(p)I

= PI p-1Ig(p) - f(p)I < w,

and so h is related to s, and h is in RE. But, according to Theorem 2.3 (4), the sets RE and . eAi are identical; therefore Z n-1,Ih(n)I < oo.

(b) follows from I.Lemma 5.1 and 99(p,s) = cp f( p,s)

cph( p,s).

11

Proof of Theorem 2.2. Assume that the multiplicative functions f and g satisfy the conditions of Theorem 2.2. Then, split f = f1 *f2, g = g1*g2,

where, using the abbreviation K = I p, the primes running through pc6:P

the set W of exceptional primes, f1 and f2 are multiplicative and f1(pk) = f(pk), if p1K, and = 0 otherwise, f2 (pk) = 0, if pIK, and = f(pk) otherwise,

and similarly for g. Then f2 is in §*, and h2 =

g2 is in AeAt by

Related Arithmetical Function,,

104

Theorem 2.1.

But, by the assumptions on the values of f and g at "exceptional prime g1_1(*) f1 = g1 , therefore fl-1() = powers pk, where p e and

h = f-'(*)*g = fl

1(*)

r g1 * f21(*) * g2 =

h2

is in 4L°JIl, and Theorem 2.2 is proved as soon as Theorem 2.3 is proved.

111.3. LEMMATA, PROOF OF THEOREM 2.3

A) WIENER-type-lemmata

NORBERT WIENER showed that the inverse of a non-vanishing, 2n-perl-

odic function with an absolutely convergent FouRIER series again has an absolutely convergent FouRIER series. An elegant proof of this result may be given via GELFAND'S theory of commutative BANACH algebras (see, for example, W. RUDIN [1966], [1973], or L. Loomis 119531). The

main part of the proof consists of the determination of the so-called maximal

ideal

space of

the

BANACH algebra of all

functions

Me"") = I cn .ein8, defined on the interval [0, 2n] using the norm IIFII = Z Icel. The maximal ideal space is the set of algebra-homomorphisms of this BANACH algebra into C, and this space is, in WIENER's case, built up precisely from the evaluation homomorphisms.

The same approach leads to the following Lemma 3.1. Denote by A the BANACH algebra of power series F(z) = Z n=1 anzn,

absolutely convergent In IzI s 1, with finite norm IIFII = Zn 1 IanI,

If F Is In A, and If F(z) $ 0 in the unit-disk {z; IzI s 1), then the power-series for the function z H 1/F(z)

111.3. Lemmata, Proof of Theorem 2.3

is in

.

105

again.

For a proof see, for example, W. RUDIN 11966], L. LooMis 11953]. The corresponding theorem for DIRICHLET series is more difficult to prove, the reason being that the maximal ideal space of the corresponding BANACH algebra contains many more functions. Lemma 3.2. The DIRICHLET series

En

where En=t anl < °°'

1

has an inverse

21n

where Zn=1 bnl

1

if and only If there Is some positive lower bound S IZ

n=1

>

0 for

ann-9I In the half-plane Re s z 0:

(3.1)

l

n=

>S

in Re s 2 0.

A proof for this result, using GELFAND's theory, is given in E. HEWITT and J. H. WILLIAMSON C 1957 ]; according to HEWITT and WILLIAMSON

this result can also be deduced from a paper by R. S. PHILLIPS [19511.

B) Splitting of functions f c into a convolution product Assume that a function f e is given; by the definition of the set ; it is possible to choose a constant P 0 with the properties

If(p)I<6 p, if p>Po,

f

(3.2)

1

and

I

EPk.

p>Pa k=2

f(pk)I < 3.

Define multiplicative functions fo, f1', f1", f2 by prescribing their values at prime-powers in the following way.

(f(p))k, if p > P0, (3.3)

0

,

if

p

sP o.

The function fo is completely multiplicative. The function defined next, f1' , is 2-multiplicative and inverse to f with respect to convolution:

Related Arithmetical Functions

106

f1' (p k )

(3.4)

- f(p), if k= land p > Po, =

otherwise.

0

The "tail" of f is defined as follows: fl..(pk)

(3.5)

f(pk), if p > po,

_

j

0

,

if

p s P0'

,

If

p > P0,

if

p

Finally, the "head" of f is

0

f2 (pk) =

(3.6)

We define f1 = f'1 * f1"

.

f(p

PO .

Looking at the generating DIRICHLET series,

it is obvious that f = f0 * fl * f).. * f2, and

f0-1(*) = f1,

.

the second assertion can also be seen from the relation

h-1(*)

= µ' h, which is true for completely multiplicative functions (see Exercise 1,8). C) The Main Lemma

then the following assertions are true: Lemma 3.3. If f, g (a) f*gE If f and g are in §.w, the same is true for f * g. (a*) (b) f0 ;'", and f -1(*) E*. .

E

(c)

f1" are in §, f1 is In .pI1'I1 In fact f1', fl" are In I ,P4L'It, where f1 = f1' * fl"

f1',

(f)

where f1 = f1' * f1".

f1-1(*)

(d) (e)

0

E

f2 E .a4L°AZ. If, for every

prime p s P0, Epf(p,s) * 0 in the half-plane

Re s Z 1, then f -1(") Is In 46AI.

Proof. Recall that §, § , p f(p,s) and AeAl are defined in (1.2) to (1.5). (a) For the moment we write w = f g. Then w(p) = f(p) + g(p), and so Ep-2'

If(p)I2+Ig(p)I2}
P

P

because f and g are In 0. If k 2 2, the definition of convolution yields w(pk) =

ZZ

r,tzO, r+t=k

f(pr)

.

g(pt).

III.3 Lemmata, Proof of Theorem 2.3

107

Thus the second sum in the definition of the set ; may be estimated by

y p

k 2 p k . lw(pk)I

E

5

s

E1+ ...

pr

P

r+t a 2

If(pr)I

p-

Ig(pt)I

2: 4'

+

with conditions of summation

prime, r22, t22

1, t22 prime, t=0or1,r22 prime, r = 0 or r = prime, r = t = The convergence of 2:P

-7

tz2

in

Y-If

in

2'

in in

1

p-t Ig(pt)I implies

3' Y_

4'

the

boundedness of

Etz2 p-t Ig(pt)I, and by the boundedness of EP p-t If(pt)I the If(p)12 is convergent. Since p-2 convergent, we obtain is sum 2:1 f(p) = Ce(p), and g e § implies the boundedness of Y_ 2; similarly, Z3 is bounded. Finally, the convergence of 7, 4 comes from the CAUCHYSCHWARZ inequality.

(az) The equality w = f * g implies the relation £(w,s) = O(f,s)

'

£(g,s)

for the generating DIRICHLET series, and YW(p,s) = Pf(p,s) 9 g(p,s)

for the factors of the EULER products, and thus (a*) Is clear. (b) The factors pfe(p,s) of the EULER product are given by (1 -

resp. p s Pot and therefore these are * 0 in Re s z I by the choice of P0. Thus fo E y°`. It is easy to check that fi and p

so

fo-ic*) = f1'

;° obviously implies that fi (c) f ;. and f1" a ;, and thus f1=f1'* ft" is in ;, by (a). The values of f1 at prime-powers are E

t:

f(pk) - f(pk-1)

'

f(p), if p > Po and k Z

f1(pk) =

t

0

Therefore, we obtain

otherwise.

2,

Related Arithmetical Functions

108

(3.7)

2: s PO
Z n-i,If1(n)I nsN

If(pk-1 ). f(p)I)}.

The assumption f c § Immediately Implies the convergence of the product on the right-hand side of (3.7). Therefore, f1 E AL°lll. Finally, the sum

SP =E ka2 is (if p > P0) SP s p-2

If(p)12 + (1 + p

pk

If(pk)I s z

by our assumptions on P0. Therefore, in the half-plane Re s z 1 I9f (p,s)I 2 1 - S 1

and so f1

P

2 2,

E

(e) The generating DIRICHLET series .V(f2,s) for f2 is a product of finitely many, in IzI s p-1 absolutely convergent power series,

Z f(pk)

zk, where z = p-s, Re s Z 1, and

kaO

psP0.

Therefore, ,(f2,s) is absolutely convergent in Re s z 1, particularly at the point s = 1, and so f2 is in .r4 'fti. (f) Under the additional assumption q f(p,s) $ 0 in Re s 2 1, for every p s Po, all the power series 21 kkO f(pk) zk in (e) are invertible with absolutely convergent power series expansion by Lemma 3.1. Therefore, the DIRICHLET series for 1/Z)(f2,s) is absolutely convergent in Re s 2 1,

and f2 1(*) is in AM M .

(d) This part is based on the rather difficult Lemma 3.2 and thus may be considered as the most difficult assertion of Lemma 3.3. The function 1 - x - exp(-2x)

is0forx=0,is2-e1>0at x=2, and has a unique local maximum at 2'-In 2 = 0.346... ; therefore (see Figure 111.2) 0,1

0,2

0,3

0.4

0,5

Figure 111.2

0.6

0,7

I

Using this inequality, the multiplicativity of f1 and the values fI(pk) given in (c), we arrive at the lower estimate

111.3. Lemmata, Proof of Theorem 2.3

p>P

{I -Z

Pk'(If(Pk)I+If(pk-1

k22

exp { - 2

>

109

S p> PO

P

}.

for all s in Re s z 1. Making use of the fact that f is in ; (and using the assumption If(p)I/p < 1/6 as long as p > Po), we obtain s

S

P>P0

P

P-2 If(p)I2 +

E

P> P0

Z Pk

L 6

p>P0 k22

if(pk)I 5 Y1

with some constant Y1, and so

n-' - ft(n) Iz8 =exp(-2Yt) inRes> 1.

IL.

An

application of Lemma 3.2 now gives fl-100C

the

desired

conclusion

,46m.

Proof of Theorem 2.3. Assertions (1) and (2) are already proved (see Lemma 3.3, (a) and (a*)). For (4), the assumption h c A1?A implies E p-1 Ih(p)I S Zn 1 n-t.Ih(n)I

<

P

therefore h(p) = O(p) and

L P-2 ,Ih(p)I2 = O { E p-1 Ih(p)I } < P

co.

P

Moreover,

Ep k22 Ep and so h

E

k Ih(pk)I

s En

n-1.Ih(n)I 1

< co,

g. The relation Z p-1 Ih(p) - s(p)I = Z p-' 'Ih(p)I < ao P

P

shows that h ands are related, therefore h E Re and AeA c Re. On the other hand, if h is in RE, then

E p-1 Ih(p)I < co, and E E P k Ih(pk)I < P kx2

P

these relations imply

IT {i+ P1

psN

Ih(p)I + kZz P k Ih(pk)I } = O(1),

therefore Re c ,em. Finally, the proof of (3) is based on some of the assertions of Lemma

Related Arithmetical Functions

110

3.3. The multiplicative decomposition f = fo* f1 * f2 gives

f-t(*) = f -i(*) * f -1(*) * f -1(*) 0 2 1

According to Lemma 3.3 (d) and (f) the convolution inverses

f2

f1-t(*)

and

t(*)

are in .r4L'A; this set is equal to RE by the, already proven, assertion (4) of Theorem 2.3, and RE c ;. On behalf of (b) of Lemma 3.3 the function f0-1(*) Is in ;* c ;, and from (a) we deduce f-t(*) f

Using (c) and (e), we see that the two DIRICHLET series

£(f1,s) = En n-s,ft(n) and D(f2,s) 1

are absolutely convergent in Re s 2 1; therefore the inverse DIRICHLET

series D(f1 1(*), s) and .(f21(*), s) cannot have any zeros in Re s z 1, is closed with respect and so fl 1(*) and f21(*) are in °*. The set to convolution. Therefore f-t(*) = f0-1(*)

*

* f2 1(*)

f1-1(*)

is in

11

III.4. APPLICATIONS

Some applications of the main theorem (Theorem 2.1) are given in this section which specify the vague remarks in the footnote at the beginning of this chapter. The methods used are well-known (see, for example, the proof of A. WINTNER's theorem in 11.2) and are based on the representation

g = f *h, with a rather "small" function h. The method is thus a linear method and does not apply aptly to problems such as the existence of higher moments.

For the definitions of the sets to (1.2), (1.3), (2.3), (2.4), and (1.4).

;p, ;,* and of cpf(p,s) we refer

111.3. Lemmata, Proof of Theorem 2.3

111

E§.p andg E § p denote mulTheorem 4.1.Let tiplicative arithmetical functions which are (i-related, so that I p-f3.l f(p) - g(p)

I

< co

.

P

Assume that, as x -) co, the following asymptotic formula holds: Z f(n) = A nsx

x" + R(x), x

co,

with the remainder term R(x) = C7( xa) Cresp. R(x) = o(xa) I

Then, for x -4 oo,

Z g(n) = A*

(4.3)

nsx

x' + R*(x),

where R*(x) = O( xa ) C resp. R*(x) = o( xp ) ],

and where the constant A is given by *

(4.4)

A

1

=

A

(f-1(

g)(n) = A U

n

A TI ( p

EP (PAY)

f (p, Y)

p

+

+f

f ipi p

P

2

+...)

x

(1

+

K(T P

)

2 +

P

2 -y

Remark. Using Theorem 2.2, the assertions of Theorem 4.1 remain true if the assumption "f E ;*" is weakened to "f E jy°, and Epf(p,s) $ 0 in Re s 2 I outside some finite excep-

tional set Of of primes, and f(pk) = g(pk) for every prime in Of and every k." The following result from H. DELANGE [1961a1 is a simple corollary; the crucial assumption f e ;* is easily obtained from the conditions on the values of f at powers of the prime 2. Corollary 4.2. Let f, g be multiplicative functions satisfying Jfl s 1, IgI s 1, which are related (that Is, Z p-'-If(p) - g(p)1 < co). If the mean-value P

M(f) exists, then the mean-value M(g) also exists, provided that g(2k) = f(2k) for every k in the exceptional case, that If(2)I = 1, and that f(2k) = (-1)k-1 { f(2) }k for every k 2 2.

Related Arithmetical Functions

112

Proof of Theorem 4.1. According to Corollary 2.5, the multiplicative function h = f-1(*)* g is in .s4f'ftt ; we obtain

h (d) f( n) = E h(d) X g(n) = I nsx d n d nsx dsx msx/d f (m) in the usual way. Inserting the asymptotic formula (4.2), and then extending the summation in the first sum to I s d < w, one obtains [using R(x) = O(xa)]

I g(n) = A-xY Z d--(-h(d) + O(x" E d-7-1h(d)j) + O(xo E d-0-1h(d)J).

nsx

dsx

d>x

dz1

Enlarging the first remainder term by multiplying every summand with )''-a (which is greater than 1), we obtain the assertion the factor (d/,, of Theorem 4.1 in the case where R(x) = O(xa).

Similar calculations allow to derive the result

in

the second case

R(x) = o(x13).

Example. If f c ;* has a mean-value, then the function f = µ2 f is in and related to f. Thus it has the mean-value (see (4.4)) M(i12f) =

1+p-1.f(p)) .(Xk2o

P-k.f(pk)1-1

Corollary 4.3. Let r > 1 be an integer, and f a multiplicative function, uniformly bounded at the prime powers, and let g be a multiplicative function such that the series P-k, I g(pk) I r E ka2 Is absolutely convergent. Finally, assume that f and g are related, and that

(4.5)

(4.6)

E

p

21

P

p_'-1 f(p) - g(p) Ir

Is finite. If, for every prime p, 9fr(p,s) $ 0 In the half-plane Re s 2 1, then the existence of the mean-value M(fr) Implies the existence of M(gr).

Proof. The boundedness of f at prime-powers and the condition on Pfr(p,s) show that the power fr is in ;*. Next, having shown (from (4.6) and (4.S)) that gr is in b°., and that fr and gr are related, then Theorem 4.1 gives the assertion. Only the proofs of (4.7)

1 P-2.Ig(P)I2r P

< ao

III.4 Applications

113

and of

X p-1 (4.8)

Igr(p) - fr(P)I < ao

.

P

are not quite so obvious. Firstly, g(p) = C)(p1'r), by (4.6), and, since If(p)I s K for all primes, using Ig(p)I S 2. Ig(p)-f(p)I, if Ig(p)I 2 2 - K, (4.6) implies that the sum p-1

zp,Ig(p)1-2K

.

Ig(p)Ir <

is convergent. So, splitting the sum in (4.7) into two sums, according 0(1) in the second to Ig(P)I < 2K, and Ig(p)I 2 2K, and using sum, the convergence of (4.7) is easily shown. The relation Igr(p) - fr(P)I = Ig(p) - f(P)I

<< jg(p)

... +

+

Ig(P)r-1

.

- f(P)I

f(p)r-lI

{Ig(P)r-lI

+

.

If(P)r-111

reduces (4.8) to a proof of the convergence of

z P-1' Ig(P) -

f(P)I

P

.

Ig(P)r-1I,

which is achieved by using the same splitting process as for (4.7).

Theorem 4.4. Assume that the arithmetical functions f are related. If

In

I

E

a.* and g c

n-1.f(n)

is convergent, then the series Zn

I

n

is convergent, and n-l,g(n) 1

where

h

= En 1 n-1.h(n) Zn 1 n-1.f(n),

= g * f-1(M)

The proof, which is similar to that of Theorem 4.1, is omitted (Exercise 2).

Theorem 4.S. Let A: N -) 7L be an integer-valued additive function with the property E

p, A(p) *O

Then, for any q c 8

q

7L,

=

P-< 00

the densities lim

x -+ o

x-1 .

I

nsx, A(n)=q

1

Related Arithmetical Functions

114

exist.

This theorem may be found, for example, in J. KuBiLius 119641, p.93. It can easily be deduced from the Relationship Theorem 2.1. Consider the multiplicative function n H exp (21d a A(n) ), where « is real.

f«

This function is bounded, and thus It

and it is related to

is in

1 e*, for z p-1

exp( 2ni « A(p)) - 1

p-1

s2

I

p, A(p) * 0

P

< Co.

Therefore, there exists a multiplicative function h« 1= µ*f« ] with the properties

f« = h« * 1, and En_1

n-1

h«(n)I < oo.

I

exists for every real

According to Theorem 4.1 the mean-value M( f «. For the calculation of Sq we consider x1

nsx, A(n) = q

f lexp(2ni« (A(n)-q) ) d« nsx O

1= x

f

1

O

exp(-2nia q )

x

1. Y e2n1«A(n)da. nsx

The function « H M(f ) is bounded and LEBESGUE-measurable; using LEBESGUE's Dominated Convergence Theorem we obtain the existence of =

S

q

lim

x -> o 1

0

folexp(-2nia q ) X-1- Z e27u1«A(n)d« nsx

exp(-2nia q ) lim

X -3

x-1,

E eZ nsx

,

b0 Mn) d«,

and thus the densities 8 q do exist.

11

The calculation of these densities is laborious. We perform this calculation in the special case of a strongly additive function A(.); in this case A(pk) = A(p) does not depend on k. Firstly, by Theorem 4.1, the mean-

value of the function f: n H exp(2nia A(n) ) is 1 + P-1.( e27t1«A(p) - 1

M(f«) _ The density 8

P

q

is then

)

) _ Z n=1 n-1' (v*f«(n)).

111.5. On a Theorem of L. Lucht

S

q

=

fo exp(-2nia q)

=

f

1

O

fo

1

115

M(f.') da

exp(-2nia q

)

G

(n

exp(-2nia q

I

(

)

II (1-

e2n1«A(p))lda l

pin

ln

J/

dn

Interchanging the order of summation and Integration (this is possible by the dominated convergence theorem) we obtain Sq =

din,

(d) = q

d-1,µ2(d)

m-1,µ(m)

m,(m,d)=1 plm=: A(p)*O

d, A(d)=q, pld=> A(p)*O

= IT (1-pA(p)*O

I )

(d)

I

d-1 µ2(d)

d, A(d)=q,

IT( 1-p

1)-1

pid

pld=> A(p)*O

III.5. ON A THEOREM OF L. LUCHT

Given q 2 1, denote by £q the C-vector-space of arithmetical functions f: IN - C with finite semi-norm (S.1)

IlfIlq = { Ilmsup x-1

E If(n) Iq }

1/q.

Note that fq is not closed with respect to convolution. For example, the constant function 1 n H 1 is in A1q = ?q n AI, where Al is the set of multiplicative functions. But, for any q > 1, the divisor function t = 1 s I is not in AIq. By contrast, (;,*) is a semi-group with identity element e, and (.*,*) is a group. :

L. LUCHT [1978] proved the following theorem, which may be considered

Related Arithmetical Functions

116

an important step towards Theorem 2.1. Theorem 5.1. Assume q > 1, and let the multiplicative functions f and g

In 9' be related. Assume, further, that, for every prime p, the factors cpf(p,s) of the generating DIRICHLET series for f are non-zero in the half-plane Re s z 1. Put h = f-1(*) g; then the series n-1 n-1

h(n)

Is absolutely convergent. In Re s z 1, It has the product representation n-s Z n=1

h(n) =

I

cpg(p,s) {

cpf(p,s)}-1

Theorem 5.1 easily follows from Theorem 2.1 for the following lemma.

Lemma 5.2. Denote by iil the set of multiplicative functions, and by Aq the intersection ,1q = .'eq n Al. Then, for any q > 1,

Aq c ;. Remark. The assertion is wrong if q = 1 (see Exercise 8). Proof of Lemma 5.2. Choose a real number E > 0 so small that (1+2E)/q is less than 1; this choice is possible since q > 1. Denote the conjugate index by q', q' = q/(q-1). HoLDER's inequality gives 21

k2

p-k

P

If(Pk)I =

Z

pk

2

If(p)1

p

1qE k

x {: Z 5 {Z Z If(pk)Iq p-(1+e)kl1/q p k=2 J ` p k22

p

k-(1 - q )

p k-(1

q

)q }1/q'

Using the assumption f e Eq and partial summation, resp. the inequality

(1 - q )

q' > 2, both of the double series on the right-hand-side of the above formula are convergent.

The assumption f e ,q implies If(n)I s Y ' nl/q with some constant y. Assuming q < 2 without loss of generality, we obtain Z P-2 If(p)I2 S Y P

Y

p-2+1/q If(p)I P

{ P

p-(1+0 If(P)Iq}1/q . { Z P P

and because f e gq , E > 0 and the right-hand side is finite, and so f

it/q" J

1, the expression on

,

e

§.

11

III 6. The Theorem of Saffari and Daboussi, II

117

111.6. THE THEOREM OF SAFFARI AND DABOUSSI, II

The theorem mentioned In the title was proved in Chapter II, 8. We prove it again easily In the special case of multiplicative functions. Theorem 6.1. Assume that A and B are subsets of IN with the property IN = A x B (direct product); a e A, b e

so every n e N Is representable as a product n

B,

In a unique way. (1)

If

I

b-1

beB

is convergent, then the density

S(A) =

lim

Z

x-1

x - oo

b-1 } 1

1 = {

acA, asx

bcB

exists.

(2) If

b--1 be

eB

Is divergent, then the density S(A) is equal to zero.

The representability of the semi-group (N,

)

as a direct product

IN = A x B is equivalent to the possibility of decomposing the constant function I as a convolution product I = a * p of the characteristic functions a, (i of the sets A, resp. B.

In the special case where a and

are multiplicative, it is easy to 1(*) deduce Theorem 6.1 (1) from Theorem 2.1. First a = * (i next the constant function I is in a", a is in , and a and I are related: 1 = 1(p) = a(p) + P(p) for primes p implies (i

1

1 - a(p) P

I

= E P-1'( 1 - a(p) P

ZP P

1

R(P)

I0-1(m)(n)I and so a = I * n-1 where < co. Theorem 4.1 and the relation y (p,l) yJ3 (p,1) = 91(p,l) give the assertion. 3-1(*),

These remarks may be considered as a hint that there might be a more general version of Theorem 2.1 in which the assumption of multiplicatlvity can be weakened.

Related Arithmetical Functions

118

111.7. APPLICATION TO ALMOST-PERIODIC FUNCTIONS

Denote by ,al [resp. B1 the C-vector-space of linear combinations of exponential sums n H ea(n) = exp ( 2ni a n ), a real [resp. a c Q], and

denote by .2 the C-vector-space of linear combinations of RAMANwAN sums

cr: n H E dl(n,r) Using the semi-norm

(r/d)

d

Iif II

a mod r, (a,r) = 1

ea/r(n).

{lim sup x 1 .1 If(n)Iq}1/q, the spaces

=

q

=

x -- >m

nsx

- closure of .al [q-almost-periodic functions],

- closure of £ [q-limit-periodic functions],

- closure of 2 [q-almost-even functions]

may be constructed. These spaces will be studied in Chapters VI and VII in more detail. In this section we are going to prove the following result. Theorem 7.1. Assume that the multiplicative arithmetical functions f and that g E and g are related, that f E (i)

(ii)

(iii) (iv)

if f Al, then g E .al if f E D then g c D if f E 31, then g E $1; if 11flll < co, then IIg111 < w. e

Remark. These assertions follow from the fact that g = f * h with a "small" function h. So the existence of such a function is also a sufficient condition for Theorem 7.1.

Proof of Theorem 7.1.

The assumptions imply g = f * h, where

(i)

Z n=1 n-1 I h(n) I < Ep (see Theorem 2.1). Given s > 0, put S = s

and choose N so large that

z n2N

n-1. Ih(n)I < s.

(1 + II f II1

)-1

111.7. Application to Almost-Periodic Functions

119

Select a finite linear combination of exponentlals near f; more exactly, f - t < e', where choose t = Z"'. ax ea with the property II

II

( 2:n=1 n-1 Ih(n)I )-1. Define the function H by

e' = s

h(n), if n s N, 0 if n > N.

H(n) =

,

The convolution e * H is in A: (eo,.

* H)(n) =

aln aSN h(d)

e,,(n/d) =

d

,,(n),

N

with the function

d a(n) =

(7.1)

10

d1'n,

if

,

e a (n/d)

,

if din.

The relation Ilsmsd exp( 2iti m n) = d if din, and = 0 otherwise, d

implies

d ,,,(n)

(7.2)

and

da

so

=

d-1

lsmsd

,

exp 2iti m d

'` +

d

J

n ),

a is a linear combination of exponentials, c A, and eC * H is in A. Using the inequality end

in

fact

11F * G1i1 5 IIGIII' En 1 n-1' IF(n)I

(7.3)

(see Exercise 6), we find IIg - t*HIII 5 11(f-t)*HII1 + IIf*(h-H)II1 5

II f - t ii

5 e'

E n=1

E n_1 n-'- IH(n)I +

II f II1

IIf 111 s

< 2e.

EN n 1 Ih(n)I

Any of the (finitely many) functions e * H is in A; therefore t * H _ 21a ea*H ) is in A, and g is in A1. (ii) If a E Q, then cad a is in £, as shown above, and so ea*H is in D

if a is rational. These remarks are sufficient to obtain a proof of (ii) by repeating the proof of (I) almost verbatim.

Related Arithmetical Functions

120

is In 2; the reason for this being that the function d(n) = cr(n/d) if din, and = 0 other(ill). The convolution cr * H : n H ZdSN

wise, is even modulo and so is a linear combination of RAMANUJAN sums. Using this observation the proof of (iii) Is performed as before.

(iv) is left as Exercise 9.

0 is multiplicative and that the series

Example. Assume that f e (7.4)

f (p) - 1

P-1

)

P

is absolutely convergent. Then f is in 21. A special case of this example is

the function n H µ2(n). Other examples are n H n

and

nH

This follows from Theorem 7.1: condition (7.4) states that f is related to the constant function 1 e §*, and this function is obviously in 2 c 21

Results for membership to $2 are not as smooth as the results of Theorem 7.1. We have to use a norm-estimate similar to estimate (7.3) used in the proof of Theorem 7.1.

Lemma 7.2. Let F and G be arithmetical functions, where F has finite (see (5.1)). Assume that G = F * h, where h satisfies the condition n-'

Z n-1

(7.5)

'

Ih(n)I < oo.

Then IIGII2 < oo. More precisely,

IIGII2 s ( zn

(7.6)

n1

'

Ih(n)I)

IIFII2

Proof. II F*h2II2 = lim sup N-1'

nN s lim sup N-1 2 nsN

s lim sup N

N-1

21

I h * F(n) 12

An

dsN tsN

I h(d) F(n/d)I

NO F(n/t)i to & nsN,

I

n-o mod [d,t]

Using the CAUCHY-SCHWARZ inequality, we obtain

III.7. Application to Almost-Periodic Functions

II F-h 1122 5 lim sup

Z

121

Z

N - oo dsN tsN

nsN

I

F(n/d)12

n-O(d)

x (N/t)-t E nsN

I F(n/t) 12

)t/2

n=O (t)

and (7.6) is proven.

Now, using the same ideas as for the proof of Theorem 7.1, we immediately obtain the following theorem.

Theorem 7.3. Suppose that f is an arithmetical function In B2, and h: IN

--) C satisfies condition (7.5). Then, again,

g = f*h is in

the function

.2

111.8. EXERCISES

1) Let f be a multiplicative function in 0*. For a fixed integer m define g(n) = f(n) if g.c.d.(n,m) = 1, and g(n) = 0 otherwise.

Prove: if f has a mean-value M(f), then the mean-value M(g) exists and is equal to M(g) = M(f)

II (i P-k

f(pk) )-1.

pim k=O

2) Assume that f is a strongly multiplicative arithmetical function, for

which the mean-value M(f) exists and is non-zero. Let g be the arithmetical function defined in Exercise 1). Prove: the mean-value M(g) exists and

M(g) = M(f)PIm n PP

1

rr (1 +

f(p) - 1

PIm

)-1

P

3) Consider the additive function A = II - w. For the densities defined in section 4, use Theorem 4.S to obtain the formulae so = rr ( 1- p-2) = 6 1<-2, P

S

q

Related Arithmetical Functions

122

I

' rIP (

Sq = fo exp (-27d a q)

1 - p-1)

(p-e2nia)-1)

(1 +

da.

4) Denote by r(n) the number of pairs (x,y) e 7L x 7L with the property that n = x2 + y2. Thus r(n) counts the number of representations of n as a sum of two squares. Then f(n) = a r(n) is multiplicative and k+1, if p = 1 mod 4, f(pk) _

if p = 3 mod 4 and k even, or if p = 2, if p = 3 mod 4 and k is odd.

1,

0,

Using 1nsx f(n) = alt x + 0( xa ), where 0 is some constant less than 1, give an asymptotic formula for Z g(n), where g = µ2 f. nsx

5) Show the existence of M(f) for the function f: n H

and

give a formula for this mean-value.

6) Prove inequality (7.3). Hint: without loss of generality, Z n-1 Is convergent. Use y-' Z IG(m)I 5 IIGIII + E, if y is large.

IF(n)I

rnsy

The function d

7) Calculate the coefficients ak in d = Zkldr is defined in section 7). 8)

Define an infinite set P of primes p1, p2, ... by p1 = 2, Pn+1 z 2-p I p2 " ' pn, and define a multiplicative function by f(Pk)

Let f

E

ifpEP,k=1,

0

otherwise.

Hints: x-1 E f(n) 5 2, but I (p-1 f(p) )2 = 00.

Prove: f E X11, f 9)

p

nsx

§* and g e§

p

be related functions, g

=

f * h,

where

Y°° n-1' Ih(n)I < co. Prove: n=1

a) if II f II1 < oo, then Ihg II1 = If II1' E n=1

n-'- Ih(n)I = Ilf II

TI zk=o

p-k

Ih(pk) I;

b) if a e IR and if the FouRIER-coefficient f (n a) exists for every n e IN, then g(a) _

n

1

n-1

10) Let f be multiplicative and q z I p-1 f(p) I2,

p

k2 P

h(n) 1.

(not).

If the series

P-1. I

p

p k I f(pk)Iq are convergent, then

If(P)Iq-1 11 f IIq

< Eb.

123

Chapter IV

Uniformly Almost-Periodic Arithmetical Functions

Abstract. This chapter deals with completions 2" c D" c A" of the vector-spaces . c D c A of even and periodic functions [and of linear combinations of exponential functions n H exp(2nia - n)] with respect to the supremum-norm 1I f II" = sup.,, If(n)l. The fundamental properties of these spaces are proved, and additive and multiplicative arithmetical

functions in the space Bu are characterized. Then some topics from probability theory are discussed and applied in order to study limit distributions of real-valued functions in 2". The maximal ideal spaces from GELFAND's theory of commutative BANACH spaces, A$ and ham, are given for 2" and for D" which are isomorphic to spaces of continuous functions defined on the compact sets A. and t . Some properties

of arithmetical functions In St' and D" can then be derived from this knowledge and standard theorems of analysis (for example, the STONEWEIERSTRASS or the TIETzE theorem). Finally, a theory of integra-

tion is developed in these spaces and applied to the calculation of some mean-values.

Uniformly Almost-Periodic Arithmetical Functions

124

IV.1. EVEN AND PERIODIC ARITHMETICAL FUNCTIONS

The C-vector-spaces Sr of r-even and 2r of r-perlodlc arithmetical functions are defined in I. 3. The pointwise product converts these vectorspaces into C-algebras. The C-vector-spaces of all even, resp. all periodic, functions will be denoted by 00

OD

r r e s p. 0 =r=1U D r

21= U

(1.1)

S,

r=1

21 and D are C-algebras. If f is even mod k then it is even mod also. Thus 2gcd[r,,r, ] c $r, U Br.

Blcm(r,,rs)*

, resp. Dr' may be described as vector-spaces with some natural bases. In fact,

The spaces .71

(1.2)

D r = Lines 1exp(27ri

r

n), a =1,

2,

rI

is the vector-space of linear combinations of exponential-functions (1.3)

ea r: n H exp(2iti r n), where a runs from 1 to r.

All these functions are r-periodic and thus in 2)r, and every r-periodic function is a linear combination of the functions (1.3), as may be seen from the linear independence of the functions (1.3) [which may be deduced from the orthogonality relations] and the fact that the dimension

of 8r is exactly r. From (1.2), again, it is clear that Dr is an algebra (a pointwise product of r-periodic functions is an r-periodic function).

The space of r-even functions is a vector-space of linear combinations of RAMANUJAN sums, (1.4)

$r = Line [cd, dir].

Obviously, since cd is even mod d and thus even mod r, all these linear combinations are in Ii r . Next, the dimension of 2 r over C is t(r) [for example, the functions gd, dir, gd(n) = I if gcd(n,r) = d, and gd(n) = 0 otherwise, are a basis for But the t(r) RAMANUJAN sums cd, where dir, are linearly independent, as follows from the orthogonality relations Sr].

IV.t. Even and Periodic Arithmetical Functions

125

(see 1.3), and (1.4) is proved. If f is an r-even arithmetical function,

then it is representable by the basis [c d' dir ] in the form f

(1.5)

cd

dEr ad(f)

with the "RAMANWAN-[FoURIER]-coefficients"

ad(f) _

p(d) cp(d)

)-1

)-1

r-1

2]

'

lspsr f(p) cd(p)

.

Formula (1.6) easily follows from the orthogonality relations for RAMANUJAN sums, already proved in Chapter 1 (3.3). The vector-space . of even functions is actually an algebra: the product of an r-even and a t-even function is an function.

Another proof of this result follows from the fact that ,Sr is the space of linear combinations of the functions cd, dir. This result is combined with the multiplicativity of the RAMANUJAN sums, considered as a func-

tion of the index (see I, Theorem 3.1), and with the relations C

(1.7')

and (1.7") c

pk

Cpk = Y(pk)

c

1

+c

p

+

...

+c

p.

,) + (pk - 2 pk-1)

These relations may be checked pointwise, using we obtain the following theorem.

1

Cpk

(3.4). Summarizing,

Theorem 1.1. The vector-space 2 of all even functions is equal to the space of all linear-combinations of RAMANUJAN sums. Thus

, = LinC [ cr, r =

(1.8)

1,

2,

... 1,

and the vector-space £ of all periodic arithmetical functions is equal to (1.9)

.V = LinC [

=

1 I ]]..

We mention that another proof for Theorem 1.1 , which is applicable in much more general situations, is given in SCHWARZ & SPILKER [19711,

Uniformly Almost-Periodic Arithmetical Functions

126

the main idea being to use the WEIERSTRASS - STONE Approximation Theorem.

Finally, we define the C-vector-space (in fact, again, A is an algebra) A = Lin, [ e", a E Qt mod 7L ]

(1.10)

of complex linear combinations of the functions ea n H exp :

The mean-value M(f) for functions in A exists, and f e A has the FOURIER expansion E

(1.11)

ea .

-oe

oe a Qt/Z

In general, arithmetical functions are neither even nor periodic. So the

spaces defined up to now are (by far) too small. In order to enlarge the spaces 8 c D c A we use the supremum norm IlfIlu = sup If(n)I,

(1.12)

nc

where f is a (bounded) arithmetical function. Obviously, I1.llu is a norm with the usual properties (i)

IlfIlu 2 0, and IlfIlu = 0 if and only if f = 0,

(ii)

II

IXI

.

IlfIlu for every complex A,

Ilf + ghIu s IlfIlu +

(iii)

ugh lu.

Moreover, in addition, (iv)

Ilf

IIgIIu,

IIf2IIu = IIfhIu2,

(v)

and there is an

gllu 5 IlfIlu

involution on R, B, A, namely the complex conjugation,

with the properties

(I)

= f + g,

(f+g)

(U)

a

=g f,

(ill) (io)

f,

(f

)

In addition, this involution satisfies

= f.

Iy.1. Even and Periodic Arithmetical Functions

(o)

II

II

=

127

IIfIIZ,

and so (after proving that $, 2), A are BANACH algebras: we shall do this in the next section), we know that 8, 2), A are B*-algebras (for the definition see RuDIN [19731, or the Appendix A.6). Working in the vector-space of all II. IIu-bounded arithmetical functions,

we construct the desired enlargements of our spaces by forming the closures of 2, 2), and A with respect to the II. IlLL norm, u

(1.13i)

=

11.11

u

-closure of 2

(the vector-space of uniformly almost-even functions), Du = II.II-closure of 2)

(1.13ii)

(the vector-space of uniformly limit-periodic functions), and Au

(1.1311i)

=

u

-closure of A

( the vector-space of uniformly almost-periodic functions).

Since A, 2,

2) are C-algebras, the $u , Du are again C-alspaces gebras. Figure IV.1 shows inclusion relations between the spaces defined up to now. Au,

Zr

r I

If

dir.

'Vd

Figure IV.1. Moreover, $u * 2)u

* Au ( R * 2 $ A is obvious ). The non-principal character X mod 4, X(n) = if n= I mod 4, X(n) = -1 if n = 3 mod 4 and X(n) = 0 otherwise, is periodic and so Is in 0, but not in 22u: assume, that X is an M-even function near X; there exist primes p a 3 mod 4 not dividing M, so gcd(M,p) = 1, X(p) = X(1), but X(p) is near 1

-1 and X(1) near 1, a contradiction.

If a is irrational, the function a is in A, but not in Du. Otherwise, choose g = X. mod r near e0' with respect to 11.IIu (where r is sufficiently large). If n = r !

m, m = 1,2,..., then

Uniformly Almost-Periodic Arithmetical Functions

128

g(n) =

a mod r

(3(a/r)

is constant, but the values of aa(n) are dense on the unit circle (the sequence {(a r! m), m = 1, 2, ... } is uniformly distributed mod 1; ,

see 11.7).

As illustration of the rather regular behaviour of functions from the spaces defined above, Figures IV.2, IV.3 and IV.4 show ad hoc constructed functions in .SLL (one multiplicative, one additive) and in Du (multipli-

cative) in the range between I and 198 Iresp. 2981. The first function is strongly

multiplicative, with values f(2)

=

0.3,

f(S)

=

0,

and

X(p).p-3/2 for primes p * 2, 5; the character X mod 5 is defined by X(2) = X(3) = - 1, X(1) = X(4) = 1, X(S) = 0.

f(p)

=

1

1.0

so

too

F I g u r e IV.2

iso

The function given in Figure IV.3 is strongly additive. Its values at the primes p>2 are f(p) = with the same character as in Figure IV.2, and the value at p = 2 is f(2) = 1. 1.0

-1

O. S

-1

0

-0. 2

Ir

II 50

I m

i

I

200

100

300

Figure IV.3

The final example, given graphically in Figure IV.4, is a strongly multipli-

Iy.l. Even and Periodic Arithmetical Functions

129

1 - p-3i2

cative function in emu, defined at the primes by f(p) =

if p>2,and f(2)=-2. 0.5 0

-o.s -I

H

0

u fl

50

iii

11

11

11

200

100

300

Figure IV.4

There is a fascinating interplay between evenness and periodicity, on the one hand, and additivity or multiplicativity, on the other. We give some simple examples below.

Proposition 1.2. If F Is additive and q-periodic, then the following hold: F(C) = 0, if gcd(C,q) = 1. (i) (ii)

F(pk) * 0 is possible at most for primes p dividing q.

(iii)

Put q = IT

(iv)

F Is q-even.

pt",. If (i > at, then F(pt") = F( pro'')'

1stsT

Proof. (i) follows from F(q) = F(q 0 = F(q) + F($); (i) implies (ii); follows from Z F( pta,) = F(q) = 1stsT

F(p1P-OC,

F(Ptot,

q) _ 2125tST

)

+ F(p1a).

(iv) If n = IT poe, then, using (ii) and (iii),

F(n) = 2]plq F( pP°) = Zplq F(

pmin(PP'cx,))

= F(gcd(n,q)).

O

Generalization. If F Is q-periodic and e-nearly-additive (this means: if gcd(n,m) = 1, then F(n) - F(m)1 < s ), then (i') (ill)

if gcd(C,q) = 1, If R Z at, then F(pto) - F(pt ' )I < 2s.

IF(8)I

<

s,

I

Uniformly Almost-Periodic Arithmetical Functions

130

Proposition 1.3. If F Is q-periodic and multiplicative, and If F(q) * 0, then F(s) = 1, If gcd(2,q) = 1. (i) F(pp) = F(p°), if p°` q and (i z a. (ii) II

(iii) If the condition F(q) * 0 is weakened to: there Is some primepower q°C for which F(q°`) * 0, then F(pk) = Fk(p) for every prime p * q not dividing q, and k = 1,2,... .

Proof of (iii).

F(q°C)-F(pk+1)

= F(qm' pk+1 + pk-q) = F( pk.(q°`p + q)) F(

=

F(pk)'F(q°L)+(p)

Proposition 1.4. If f Is multiplicative and q-periodic, then f(pk) = 0 for some k is possible for at most finitely many primes. Proof. Assume there are infinitely many prime-powers prkr for which f(prk,) = 0. Without loss of generality, we may assume that these primepowers are coprime with q. Denote by ar the residue-class of prk, mod q; these residue-classes are in (7L/q7L) . At least two of the residueclasses 'T15rsR ar are equal, so there are integers R and S, for which

n ar RS

1 mod q.

Then

1=f(1)=f(R
r

a contradiction.

In the case of completely multiplicative functions the same argument gives:

If f Is completely multiplicative and q-periodic, then (i) f(p) = 0 is possible only if plq. (ii) If(n)I a {0,1} for any Integer n, f(n) Is zero or a root of unity, (iii) if q > 0 is the least period of f, then f(p) = 0 for any plq. Therefore f is a DIRICHLET character mod q, where q is the least period of f. Proof of (iii). If q = q 1 q2, where f(q1) * 0, then

IV.1. Even and Periodic Arithmetical Functions

f(a)'f(g1) =

f(ag1+g2'g1)

= f(q1)

131

f(a+q2),

and so q2 is again a period of f. The next result is due to D. LEITMANN and D. WOLKE 11976]. The proof given is from SCHWARZ (Monatshefte Math. 1978). An arithmetical

function is called p-multiplicative if the relation f(p1 p2) holds for all primes p1 * p2.

=

proposition I.S. Let F: IR -) C be a periodic function with Irrational period a > 0. If the set ,P = { t e IR, F continuous at t }

Is non-empty, and If the restriction f of F to IN Is p-multiplicative, then F(P) = (0) or F(r) =(1). In particular, If F Is continuous, then F = 0 or F = I Is constant.

Remark. The example F(t) = cos(2nt) shows that the condition "a is Irrational" Is necessary.

Proof. The results needed from the theory of uniform distribution may be found, for example, in KuIPERS-NIEDERREITER [1974], Theorem 6.3,

example 6.1 and p.22. Assume there is some t e .? for which NO $ 0. Fix E, 0 < E < -IF(t)I. Let t' be an element of P. The continuity of F at t, t' implies the existence of a S > 0 so that (1.14)

IF(t) - F(x) I < E, IF(t') - F(y)I < E,

Choose a real t

if It-xI, It' - yl < 28.

It*-tl < S, such that 1, a and a- t5 are Q-linearly

,

independent. Then the 2-dimensional sequence of points { a-1q }, { a It*.q }

), q prime,

is uniformly distributed modulo 1 in the unit-square in IR2. Therefore, we find a prime q and integers m1, m2 satisfying a-Iq + m1 - a-1t' I

I

< O c-

S,

and I

O(_

I

t q+ m 2- 0(-1 t I< a-I

S.

Having fixed q, there exists a prime p $ q and an integer m for which

Uniformly Almost-Periodic Arithmetical Functions

132

m-

a

I<

S,

and so I

t

p'q +

I

< 28.

Continuity at t and a-periodicity, resp. continuity at t', give I

F(pq) - F(t) I <E, IF(q)-F(t')I <E.

And

I F(p) - F(t) I < E, because I p + ma-tI < 28, for It-t* I < 8 and (1.14). Using p-multiplicativity, we deduce

F(q) - 1

I

< 2E

IF(p)I-1

< 4E

IF(t)I-1,

and so IF(t') - 11 < ( 1 + 4 IF(t)I-1 )

Thus F(.°) = (1).

E.

11

Finally, we mention that a characterization of all multiplicative, periodic arithmetical functions was given by N. G. DE BRUUN [1943] and also by D. LEITMANN and D. WOLKE [1976]. We do not reproduce the proof here, but simply quote the result.

Theorem 1.6. A multiplicative arithmetical function f Is periodic If and

only If there exists an integer N and a DI>vcHLET character X mod N with the following properties:

If pIN and k e IN, then f(pk) = 0. (ii) If pIN, then the function k H is constant and $ 0. (iii) There are at most finitely many primes p for which I for some exponent k. (i)

rv,2. Simple Properties

133

IV.2. SIMPLE PROPERTIES

First we prove the following theorem. Theorem 2.1. The algebras Bu, Du, Au are BANACH algebras (and so are complete with respect to . IIu), and the supremum-norm has the properties (i) - (v) and (o) of section IV. 1. II

Proof. Let us prove, for example, (iii) for Bu: given some s > 0, there are functions F, G in B satisfying

E Bu and f - F IIU < s, II g - G IIu < E. Then f + g - (F + G ) 11U < s, and so (f + g) E $u. Next, Bu is an algebra: given f, g in Bu , and s > 0, there are functions F, G In B satisfying II f - F IIu < s, II g - G II u < E. Then II F G - f g IIu f, g

II

II

5 IIf - F

GIILL <

is near II g II Is In B, and so f g E Bu. ded and

II G II

u

since IIghIu and IIfIIu are boun-

property (iv) of the norm is used. F G

Concerning the completeness of, say, Bu, we assume that (Fk), k = 1, 2, ... is a I I . IIu- CAUCHY-sequence in $u. Then the values Fk(n) are a CAUCHY- sequence in (C, . ), and are therefore convergent to some complex number F(n). The function F : n H F(n) satisfies II F - Fk IIu 5 E if 1

1

k 2 k0(s), so F is the II.IIu - limit of the sequence Fk. Finally F is in Bu because F is near Fk if k is large, and Fk is near some fk in B. O Theorem 2.2. Assume f, g E Bu [resp. E Du, resp. E Au ]. Then the functions

Re f, Im f,

IfL,

are again in Bu [resp. Du, resp. ,v4uJ. If f, g are real-valued, then

f+ = max (0, f) and f

min (0, f),

and, more generally max( f,g) and min( f,g

)

are again in Bu [resp. Du, resp. 4u ]. The shifted functions (with positive integers a, b) fa : n H f(n+a), and f b,a: n H f(bn + a)

Uniformly Almost-Periodic Arithmetical Functions

134

are In Du 1f f is 1n 2 u or D u, and in 8 u if f Is in,4 u. Proof. The result for Re f and Im f is obvious. If f is near p e $ [resp, e], then Ifl - lpl 15 f-p I, so Ifi is near 191 and IyI is even [resp periodic] and so again is in 3 [resp. £]. I

I

If f is in Au and near (p in A, then there seems to be no easily accessible structural property 1) which is obviously true for 191. But the WEIERSTRASS Theorem (see Appendix, Theorem A.1.1) shows that 191 is in ,v4u: for 191 is bounded, say Ipl s M. Given e > 0, there is, by the WEIER-

STRASS Theorem, a polynomial P(X) with real coefficients, satisfying IP(x)- lxi < e in -M s x s M. A being an algebra, the function P(p) Is in A, and ll P(w) - Iwl Ilu < E, and so 191 a Au, Therefore, Ifl Is in Au. The formulae max (f,g) = 2(f +g)+ 2''If-gl, min(f,g) = 1(f + g) show the assertions concerning max(f,g) and min(f,g).

If 9 is in D, resp. A, then the shifted function Ya is clearly in £, resp. A (similarly, cpb,a is in £, resp..); and 9. is near fa if 9 is near f. Theorem 2.3. If f is in Au then the mean-value M(f) exists. Moreover, the FOURIER coefficients

?(a) = and the RAMANUJAN coefficients

ar(f) = {p(r))-1 . exist.

Proof. Without loss of generality, let f e > 0, there exists a function

FE

e

Au be real-valued. Given

.A with the property

F(n) - e < f(n) < F(n) + e

for every n e N. The mean-value M(F) exists, therefore the difference

of the upper and lower mean-value of f, is 1)

Of course, p Is (see, for example, CORDLINEANLI 11968] ) almost-peri-

odic, and so there are c-translation numbers for cp; these are also c-translation numbers for 191, and so jp1 Is In Au.

Iy.2. Simple Properties

135

IM_(f) - M _(f)1 s e,

and so M(f) exists. If f E A", then

are also in A", and

and

thus the assertions about the FouRIBR and RAMANWAN coefficients are clear.

Theorem 2.4. Let f E A", and let X c C be a compact set with the following property: there is some S > 0 such that UN B(f(n), S) c X. B(f(n), S) denotes the ball with radius S around f(n). Assume that : .2' --3 C is LIPSCHITZ-con tin uous;

so there is a constant L with the property 14(z) - 4(z')I s L

z - z' I, if z, z' E.X.

I

Then the composed function N - C

4rf

is again in A". The same result is valid In .$".

Proof. Let a be less than 8. If F in A is near f, f - F II" < e, then the values of f and F are in X; by the LIPSCHITZ-continuity, III0f - 40F11" s L e. We have to show that °F is in A". According to the complex 11

version of the WEIERSTRASS Approximation Theorem, there is a polynomial P(z,z) with complex coefficients, so that

14(z) - P(z,z) Thus

4(F(n)) - P(F(n),F(n))

I

I

<

I

< e, if z E X. e

for any

n

E

W;

the function

n H P(F(n),F(n)) is in A, and so 4°f is in A". Corollary 2.5. (1) If f E A", then

eix-f

E A" for every complex constant X. z S, where 8 > 0, then 1/f is in A". (2) If f c A" and Ifl

(3)

I f f is in A", (z

IN>

8,

where S > 0, and If there Is an angle

E C, Iarg(z) -a1 > 8) free of values of f, then log(f) is in A".

Theorem 2.4 is a special case of the next, more general, theorem.

Theorem 2.6. Let f c A" (resp. f c $"), and, for y > 0,

Uniformly Almost-Periodic Arithmetical Functions

136

KY = { z E C: In E IN with the property If(n) - zI < y }. Then, for every continuous function 4): KY - C, the composed function i{r°f

:

IN -) C

is again in Au (resp. in 2u). Proof. The function f is bounded, therefore the closure KY/2 is compact and 4), restricted to KY/2, is uniformly continuous. Given s > 0, there is a 8, 0 < 8 < Zy such that I4)(z) - c)(z')I < s for all z, z' E KY/2' 1z-z'1 < S.

Choose a function F in A (resp. in .) near f, If - F IIu < S. Then Il4of-40FilusE.

If f E Su, F E $, then 40F E 2, and 4rof E $u. If f E Au, F

then 94u by the WEIERSTRAss Approximation Theorem (as in the c .F proof of Theorem 2.4). Therefore, 4rof is in Au. E

,v4,

The next result contains a characterization of the additive functions of to Su. Theorem 2.7.

(1)

If f is in Au and Is additive, then scup

(1)

If f is in 21 u, then lim

(ii)

k -> m

I

f(pk)

< oo.

P

f(pk) exists for every prime.

(2) If f is additive and If relations (1) and (li) are true, then f is In Su.

(3) If f is in Du and is additive, then (ii) is true, (4) Therefore, the Intersection of the vector-space of additive functions with Du is equal to the intersection of this space with Bu. Proof. (l.i) Without loss of generality, f is real-valued; f is uniformly bounded, and so IZ f(pk)I s Ilfllu, summed over any finite set of primepowers for which f(pk) z 0 (and the same is true for every finite set of prime-powers for which f(pk) < 0 ). These remarks imply

IV.2 Simple Properties

137

21 sup I f(pk)I s 2 p

k

IIfIIu+ 1.

(l.ii) The values f(pk) are bounded, so there is a subsequence k1 < k2 < Ln-1, if kr z K1(n). , for which f(pk,) is convergent, I

Choose Fn a . 7 1 near f,

1 1F

n - f IIu < n-if k 2 K2(n) is large, then the

values Fn(pk) are constant, and thus I L - f(Pk)I s I L - f(pk,) I

+

f(pk,) - Fn(Pk,)I + IF n( Pk) - f (Pk)I <

I

if k, kr z max ( K1(n), K2(n) ), and (ii) is proved.

.

(2) Assume f is additive and satisfies (i) and (ii), limk---, f(pk) = g(p). Choose E > 0. There are constants P 0 and k0 (depending on E), so that P

Po kP

If(Pk)I < E, and If(pk) - g(P)I <

if k z ko. Put K = rI pko and define a K-even function F by PSPO

F(n) = f(gcd(n,K)).

Write n = TI pvp(n) = n'- n", where n' contains those prime-factors of n which are s Po, and n" contains the "large" prime-factors p > Po. Then I f (n") < E by the choice of Po ( and by additivity ). Decompose n' = n1 n2 where n1' contains the primes p with vp(n) s ko and n2" contains the others. Then We aim at II f - F IIu <

,

F(n') = f(n1') +

P

nz

f(Pk°)

and so

IF(n) - f(n)I s If(n")I + pn If( Pv'(n) - f(pko)I < E +

2E.

z

(3) Let p be a prime, IIf-FIIu < E. If IF(

pk)I

If

E

>

0. Choose a function F

e

£, r-periodic,

then

s IF(Pk)-f(Pk)I + IF(r) -f(r)I + (pk)I < 4E for every k if p.' r. If plr, say pk IIr, then, similarly, IF(Pk) - F(Pko)I < 4e,

therefore

If(pk) - f(pko)I < 6 E for every k z ko

Uniformly Almost-Periodic Arithmetical Functions

138

If(pk) - f(pe)I < 12E if k,E s ko.

So k H f(pk) is a CAUCHY sequence, which proves (ii).

(4) follows from (1) - (3).

11

Theorem 2.8. Assume that f is in Bu. C[X] Is a polynomial with complex coefficients, then (1) If P E

P.f E Z)u.

(2) If P E 7L[X] is a polynomial with Integer coefficients and P > 0, then Proof. Du and $u are algebras, and so (1) is clear for Du and 5u. Approx-

imating f by a finite linear combination of functions e./r' it is easy to reduce assertion (2) to the problem of showing that n H e./r( P(n) ) is in D'; but, due to P(n+r) = P(n) mod r this function is periodic and so

itisin2.. Finally, we give the following uniqueness theorem. Theorem 2.9. Assume that q z 1, f E Du and II f II

=

{ lim sup X-'- E X -> co

nsx

II f II

If(n) Iq

q

= 0, where

}l/q.

Then f = 0.

Proof. Assume, on the contrary, that there is some no, for which If(no)I = 8 > 0. Choose E = there is a function F in £ near f, so that II f-F Ilu m in N)

<

E. F Is periodic with some period K. Therefore (for any E = IF(no)1 - E 2 If(no)I -2E = Zs,

2

and

II f IIq Z { Xm sup X

a contradiction.

X-1 .

n

nsx,ne mod K If(n)Iq }l'q.

11

Remark. Other uniqueness theorems are proved in VI, Theorems 1.5, 1.6.

jy.3. Limit Distributions

139

IV.3. LIMIT DISTRIBUTIONS

First we have to repeat some definitions and notation from probability theory; see, for example, RENYI [19701, and the Appendix.

A function F: IR - IR is called a distribution function if (I) F is monotonically non-decreasing, (ii) F is right-continuous, so that for every x

limE o,t>o (iii) F(-oo) = limx F(+oo) = limx-->

-

e

IR

F(x),

F(x) = 0, F(x) =

1.

So, in fact, F: IR --) [0,1]. Note that the set of discontinuities of a distribution function is at most denumerable.

Examples of distribution functions are if x the function x H e(x) = J 0 ,

1

,

<0 if x 2 0.

the GAUSsian normal distribution x N F(x) =

1

X { 0., n }-I . exp{ -

-2 I'.0

(-µ)Z} dµ.

if (0, P) is a probability space, X: fl -3 IR a real-valued random variable, then (3.1)

F(u) = P(t: X(t) < u) Is the distribution function associated with X.

A sequence FN of distribution functions Is said to converge weakly [in the sense of probability theory] to F if F (x) urn N-= N

= F(x)

is true for all points of continuity of F. The limit-function F need not be a distribution function, but It is non-decreasing and bounded. In the theory of distributions the FOURIER-STIELTJES transform of a

Uniformly Almost-Periodic Arithmetical Functions

140

distribution function F is called its characteristic function f, f(t) =

(3.2)

-co

eltx dF(x).

There is an inversion formula: if a and (a+h) are points of continuity for F, then F(a+h) - F(a) =

(3.3)

lim T-. o0

fT (it)-1,(1-e-ih)

elta,f(t) dt.

,

Further properties of the characteristic function f of a distribution function are listed below: (1) f(0) = 1. (2)

If(t)I

(3)

f(-t) = f(t).

s

1.

(4) f is uniformly continuous on R. (5) If ak are non-negative real numbers, 2: isksK ak = 1, and If fk are characteristic functions, then isksK a f k is a characteristic function again; in particular, Re f is a characteristic function. k

(6)

If f is k times differentiable at 0, then the moments ak = J

of F do exist up to the order k if k is even, and up to the order (k-1) if k is odd. Continuity Theorem. If IF n) is a sequence of distribution functions with characteristic functions t H fn(t), then {Fn} converges weakly to a distribution function F If and only If (I) limn->

co

fn(t) = f(t) exists for every t

e

IR,

(ii) the limit f is continuous at 0.

Then f is the characteristic function of F. For a proof see, for example, LUKACS [1970]. The application to arithmetical functions rests on the fact that

xHFN(x)=N-l

nsN, g(n)sx

1

is a distribution function, if g is a real-valued arithmetical function; x H FN(x) is non-decreasing, FN(-co) = 0, FN(co) = 1, and is right-

IV.3. Limit Distributions

141

continuous. Its characteristic function is

eitx dFN()

fN(t) = f N

nsN

eitx d( nsN,

('

N-1

=

gW ()sx

1

l /

e it-g(n)

Theorem 3.1. Let g e Au be real-valued. Then there is a limit-distribution

F (x) (if x is a point of continuity for F

F(x) = lim

N>m N

for g.

The proof is a direct application of the continuity theorem (note the fact that it is not assumed that g is additive). If t is an arbitrary real number, then the function n H exp( itg(n) ) is in Au according to Corollary 2.5. Thus the sequence of characteristic functions fN(t) =

N-1 -

2:nsN exp( itg(n)

)

converges for n - co to the mean-value

M( n H exp(itg(n)))

NO.

The inequality eiu

-l

I

=

I

i fo e'

fu

di; I

S

I

e1E I

I

di; I

s K Jul,

if u is in the disc B(O,R), with some constant K = K(R) [if u is real or, more generally, if Im(u) 2 0, then it is possible to take K = 1], gives N1

.

nsN

( exp( itg(n) ) - 1

s

N-1

E

nsN

and so, as N tends to infinity, If(t) - f(0)I s

so that f is continuous at t = 0. An application of the continuity theorem for characteristic functions gives the assertion. The theorem given above may be extended to classes of arithmetical functions that are much larger than Au; this will be done in Chapter VI, 8 A.

Uniformly Almost-Periodic Arithmetical Functions

142

IV.4. GELFAND's THEORY: MAXIMAL IDEAL SPACES

Some notions and definitions from functional analysis are used In this section. We refer to the Appendix, A.6. The algebras Bu Bu c u are commutative BANACH algebras with identity element e = 1, and there is the "standard" involution f H f (complex conjugation) satisfying f f Ilu = f IIu2. So these spaces are II

II

B -algebras, and, according to GELFAND and NAIMARK's Theorem, these

algebras are essentially algebras of continuous functions on the [compact] maximal ideal space A. The GELFAND transform f

(4.1) 'Bu

-a e(AS)

f

resp. ^:

:

- C, f (h) = h(f) ), u --) L°(AV) resp. ^: Au -, L°(A.4)

is an isometric isomorphism in each case.

IV.4.A. The maximal ideal space A$ of 8'. a) Construction of some algebra-homomorphisms. Clearly, for any integer n e IN, the evaluations hn : f H f(n) are elements

of Ate. Next, for any prime p, and for f e Su, the limit f (pm) = liimm f(pk)

exists, as shown in Theorem 2.7, and so the functions hPm : f H f(pm) are elements of AJ9. More generally, given exponents kp, 0 s kp s co, a (complex) value f(X) can be defined for the vector

X - (kp)p prime in the following manner 2): consider the increasing sequence nr of positive integers 2)

We think of the sequence of primes being ordered according to size. An Integer n may be described as a special vector X, where at most finitely many of the kv are non-zero and none Is Infinity.

N.4. Gelfand's Theory: Maximal Ideal Spaces

T nr =Ispsr 11

min(r,k pp

)

Pp

,

143

r = 1,2, ...,

with the property nrlnr+l . Then f(X) = lim

r-4 m

f(n

r

exists3), and

hx: f H f (X)

Is an element of A$. All these functions hx are different, as can be seen by evaluating hx on suitable RAMANLI,JAN sums cqC, where q is prime.

Our goal is to prove that we obtained all the elements of A$. Before doing this, we calculate the values of hx at RAMANLJJAN sums cge for prime powers qz Obviously (giving the greatest common divisor on the right-hand-side a natural interpretation), .

hx(cge) = cqE( gcd(f pkP, qe )), and this equals

cge(q') = y(q'), cgt(g8-1) = - qt-1

(4.2) =

0,

if kq z e, if

kq = E-1,

if kq<2-1.

b) Determination of h$. Following the paper by T. MAXSEIN, W. SCHWARZ and P. SMITH 11991]

rather closely, we are going to prove Theorem 4.1. Theorem 4.1. The maximal Ideal space 0$ consists precisely of the func-

tions hx, where X runs through the set of vectors (kp)p prime'

0skpsao Proof. Assume h c A$; h being continuous, it is sufficient to know the

e. The function F is even, and so F(nr) = 13 Is constant for r z r0(e). Thus the sequence r H f(nr) is a Cauchy sequence.

3) Given e > 0, choose F c $ satisfying Ilf-FII u

<

Uniformly Almost-Periodic Arithmetical Functions

144

values of h on the subalgebra $ of $u. The RAMANLUAN sums cr, con-

sidered as functions of the index r, are multiplicative. Therefore, it is sufficient to know the values h(cgg) for prime-powers qz.

p(q'), -qE-1, 0 } if $ > 1, Since h(f) a spec(f), and spec(cge) is {p(q), -1 } if t = 1, and (1) if $ = 0, there are at most three possibilities for choosing the value h(cq ). However, not every choice is admissible. The relations {

(4.3')

Cpm'Cpe = ep(p")'cpm

,

if m > $,

and

cpZ'cp,e = p(P')'(c1 + cp +... + cppe_1) + (pe-2pe-1).cpt

(4.3")

imply (using the fact that h is an algebra-homomorphism; q denotes a prime) (a)

h(cgm) = 0, if h(cgg) = 0 and m > $,

h(cgt) * 0 and 0 s m < l;,

(b) h(cgm) = cp(gm), if

(c) h(cgt) < 0 is possible for at most one 2

(d) if

( q fixed

h(cg.,j) = 0 but h(cge) $ 0, then h(cgt) _ - qe-1 < 0

Therefore, either h(cgm) = p(qm) for any m 2 0 (define kq = case), or there exists an exponent kq such that

y(gf'), (4.4)

h(cC) =

co

.

in that

if tskq,

- qR-1, if t = kq+ 1, 0, if k > kq+ 1.

Then, for the vector X = (kq)q prime , we obtain h = ham, and so A2 is completely determined.

c) Topology. The GELFAND topology of A GELFAND transform (4.1) n

f: 0

is the weakest topology that makes every n

f(h) = h(f) continuous. So, for any prime power qz and any open set 0 in C, the sets cqg

h e A; h(cge) e 0 }

IV.4. Gelfand's Theory: Maximal Ideal Spaces

14S

are open. Therefore, using (4.4), the sets

kp arbitrary for p $ q, kq 2 Z 1,

( hx. where Z e IN, and

( hx

kp arbitrary for p $ q, kq = Z-1 }

are open. Choosing these sets as a subbasis for the topology, we see that every f is continuous. For: Given s > 0 and f, choose g = 2:1SreR satisfying IIf-gllu < ze. Assume that h e As , h = h_T, X = (kp(h)), is given. An open neighbourhood U(h) of h is defined by the condition h* a U(h) iff h*= hx,*, and kp(h*) = kP(h) for any psR.

Then h(g) = h*(g) for any h* in U(h), and so I

f (h) - f (h*)I

h(f) - h*(f) S

li

f-g

11

I

s

I

h(f) - h(g) I+

I

h*(f) - h*(g)I

+IIf - g 11u <£.

Thus, f is continuous and so the topology of A2 is completely determined. It coincides with the product topology on the space

IN * = fj [I, p,

(4.5)

P

P2' ..., p=: },

where each factor is the ALEXANDROFF-one-point-compactification of the discrete (and locally compact) space (1, p, p2, ... I.

d) Main result. For functions f in Bu, obviously 11f211" = Ilfllu2, and so we obtain from 11.12 in W. RUDIN 11966] a result already mentioned at the beginning of this section. Theorem 4.2. The Banach-algebra BU is semi-simple, and the GELFAND

transform f H f is an isometric algebra-isomorphism from ,$u onto e((12) .

Note that semi-simplicity immediately also follows from the fact that the evaluation homomorphisms hn : f'-f(n) are in As, and so the assumption f e radical(Bu ) = n kernel(h) implies f = 0. he Ap;

Uniformly Almost-Periodic Arithmetical Functions

146

Next, RUDIN [1966], section 11.20, implies the following corollary.

Corollary 4.3. If f E $ u is real-valued and if inf f(n) > 0, then there ncN exists a [real-valued] square-root g of f in 8

u.

e) Applications.

The following result is well-known and can also be derived from the WEIERSTRASS approximation theorem (see Corollary 2.5); we deduce it from our knowledge of A2.

Corollary 4.4. Assume that f

E R u. Then i / f

E S u if and only If

infnENlf(n)I Is positive.

Proof. If 1/f e $u , then this function is bounded and so Ifl is bounded from below. On the other hand, according to GELFAND's Theory (see RUDIN [1966], 18.17), i/f E 8u if for any h E A2 the value h(f) is not zero. The values h(f) are given as certain limits in section 2, and the condition Ifl z S obviously implies that all these limits are non-zero, and corollary 4.4 is proved. This corollary may be extended considerably.

Theorem 4.S. Let f c ,$u be given. If the function F is holomorphic in some region of C, Including the range f (A$) off , then the com-

posed function F,f is in L'(&) and thus is equal to some g gE

,8 u. Therefore, Fof Is in Bu again.

Except for the last sentence, this is a specialization of L. H. LooMis [1953], 24 D. Next, g = Fof implies h(g) = F(h(f)) for any h in A., and so the assertion is true if F is a polynomial [then F(h(f)) = h(F(f)]. The general case follows from this. In the case of multiplicative functions, the following results are true.

Theorem 4.6a. Let f E Ru be given. If f is multiplicative, then f(pk) = 0 Is possible for at most finitely many primes p, and the same argument gives the following stronger version of Theorem 4.6a.

Theorem 4.6b. Let f E $u be given. If S > 0 and f is multiplicative,

then there are at most finitely many primes with the property

N.4. Gelfand's Theory: Maximal Ideal Spaces

f (p k) -

11

147

> S For some k.

proof. f (hxo) = 1 where X o = (kp), kp = 0 for any p. Given E = 28, then there is some neighbourhood 14 of h with the property I f (h) - I I < E for h in R. . But this neighbourhood contains all ham, with kp arbitrary

except for finitely many primes; for these exceptional primes kp = 0 may be taken. Next, f being multiplicative,

?(h) = lim JI f(pmin(kp,L)) L-> ao psL

and this implies, by a suitable choice of the kp , and noting If (h) - 1 I < E, that If(pk )- II > E is impossible for any "non-exceptional" prime and any k. 11

IV.4.B. The maximal ideal space A.

a) Embedding of A

of D°

in rciN IT

Define, using the abbreviation Wr = exp(2ni/r), an element fre £ by fr(n) = Wr. The set of functions (4.7)

{ f* ,

1 s t s r, gcd(t,r) = 1, r = 1,2,... }

is a basis of D. A function f in D is r-periodic for some r, and so 1/f is again r-periodic and in D c DLL, if f does not assume the value zero. Therefore, spec(fr)

=

{Wr, 1 s j s r }.

If h e 0s,, then (4.8)

h(fr)= Wr(r,h)

where j(r,h) is some uniquely determined integer modulo r, depending on h. Thus we obtain a map (4.9)

cp: 0 -3 IT

defined by p(h) = ( J(r,h) )r=12

rcN

, where h and j are related by (4.8).

Obviously, cp is infective.

Examples. (1) If f is a periodic function with period M, and if H is a homomorphism in Ate, then H(f) = f(j(M,H)).

Proof. H(el/r) = er(j(r,H)). The FouRiER expansion f =

a F+eµ/M

Uniformly Almost-Periodic Arithmetical Functions

148

implies the result. (2) If g is in Du, and G is M-periodic, IIg - G 11u< e, then I H(g) - g(j (M,H)) I < 2e for every H in A... (This depends on the fact that JH(f)h s 1117114.)

(3) If hr is the evaluation homomorphism f H f(r), then j (k,h,) = r mod k for k = 1, 2, ... .

A

b) The Priifer Ring 7L

For any n e IN consider the residue class ring 7L//n-7L with discrete topology. If mmn, then there is a continuous projection (4.10)

7tm,n

a mod n ) y ( a mod m ).

7L/n.7L

The set X = IT Z/r.7 with the product topology is a compact HAUSrciN

DORFF space, and the set (4.11)

7L = { (an) E X , an E 7L/n.7 and ltm,n((Xn) = am, if mmn }

is a closed subspace of X and therefore is again compact (and HAUSDORFF). Note that IN is dense in 7L; the reason is that, given an element (ar )r in 7L, and given positive integers ri .., rN there exists an integer m c IN satisfying m = ari mod ri for I s I s N. ,

,

Since fr-s = fr It follows that j(rs,h) ° j(r,h) mod r for any h

e

A2,.

Therefore, the image of the map cp is contained in Z. A

c) Surjectivity of p: A2, -3 7L

Let some element ((xr)r in 7L be given. Our aim is to construct an algebra-homomorphism h A2, satisfying cp(h) = (ar )r. Define a linear map h: 2) -4 C on the elements of the basis of 2) by E

h(fr) = car, - °cr,

1 s k s r, gcd(k,r) = 1, r

and extend h linearly to B. Then h is multiplicative on 2): assume first that gcd(r,s) = 1; then the relation r-l-as = ( s-k +

mod rs

implies

h(f' fs) = h(fr) h(fs ). This is also true If gcd(r,s) $ 1; without loss of generality, r and s may

IV.4. Gelfand's Theory: Maximal Ideal Spaces

149

be assumed to be powers of the same prime, and then the assertion is easily checked. v h is continuous on £: given an element $ e B, 4) _ 11,,1N av-frk there exists an m e IN, for which m = (Xrv mod rv for 1 s v s N . Since

h(4)) = 4)(m), we obtain s I4)(m)I 5

and so h is continuous on D. This space being dense in Vu, h may be extended continuously to cp(h) = (ar)r=1,2,...

an

algebra-homomorphism

of

Bu,

and

'

d) Continuity of p : AZ) -> 7. . Fix ak E

I

s k s N, with the property an = am mod m if min.

Then V(a1, ..., aN)

A 7L

, ak= ak for 1 s k s N

}

is a typical basis element of the [product] topology of Z. Moreover, he cp-1(V(a1 ..., aN )) if and only if h(fk wk k for any k in 1 s k s N. This is equivalent to f k(h) = wk k, 1 s k s N, where f k is the GELFAND transform of fk.

If Llk is a neighbourhood of wkk not containing any other kth root of unity, then it follows that 1

(V(a1

N ^ -1 ..., aN )) = kn1 fk (uk

is an open set in the GELFAND topology of A.., and so 9 is continuous. A Since A., and 7L are compact Hausdorff spaces, 9 is a homeomorphism. Thus we obtain the following theorem.

Theorem 4.7. The maximal space A., Is homeomorphic with the PrUfer Ring Z, defined In (4.11).

Remark 1. The evaluation homomorphisms hn are dense in A.). Proof of Remark 1. Given H in AM, choose a neigbourhood u(H) "defined

by R"; this means that h e u(H) iff j(r,h) = j(r,H) for r in Define the integer n as j(R!, H). Then (4.12)

n = j(r,H) mod r for r = 1,2,...,R,

1

s r s R.

Uniformly Almost-Periodic Arithmetical Functions

ISO

and hn is obviously in U(H).

11

e) Arithmetical Applications

Next, we apply our knowledge of the maximal ideal space to the problem

of the characterization of additive and multiplicative functions in $u. Some of the results have already been proved In section 2 using ad hoc elementary methods from number theory. In (1943] N. G. DE BRUIN characterized multiplicative, almost-periodic arithmetical functions. Additive, almost-periodic functions were characterized by E. R. VAN KAMPEN (1940). The results are as follows.

Theorem 4.8. Assume f to be fibre-constant. Then f Is In 2 u if and only if limk , . f(pk) exists for every prime p. This result Is not true for Z)u, as the example of a character X satisfying X(p) * 0, 1 shows.

Remark 2. f is termed fibre-constant if there Is a prime q such that f(n) = f(gcd(n,q°°)) for any n. Obviously, limk--> f(pk) exists for any prime p * q trivially.

Theorem 4.9. An additive function Is In Bu if and only if (4.13)

and

lim k-o

exists for any prime

f(pk)

E sup I f(pk) I < 0

(4.14)

k

p

Theorem 4.10. A multiplicative function Is In 8u holds and If (4.IS)

E sup f(pk) p

k

I

if

and only if (4.13)

I I< OD

is true.

Remark 3. If f is in Bu then the GELFAND transform f is continuous at h., where X = (kP )P , and kq = 00, kP = 0, If p * q. All the func-

tions h.., where kP = kP = 0 for p * q, and kq = L, L sufficiently large, are near hx, , and thus the limit relation (4.13) is true.

N.4. Gelfand's Theory: Maximal Ideal Spaces

151

The proof of Theorem 4.8 now follows from the preceding remark and

the fact that for fibre-constant functions f(h) may be defined in an obvious manner using the limit relation (4.13) at q. The resulting function f is obviously continuous and so f is in $u .

For additive functions in Du we prove the following theorem. Theorem 4.11. IF f Is in Du and additive, then limk

-4 . f(pk) exists for

every prime p, and relation (4.14) is true; therefore an additive function from Du is in fact already In ,$u. proof. Given s

>

0, choose an M-periodic function F in £ satisfying

II f - F Ilu < ; E. Then F is E - nearly additive, and so, according to section 1, IF(pO)I

<

E,

if

p does not divide M, and IF(pp) -

F(p°C)I

2E

<

If

(i > a and p°`IIM. This implies that k H f(pk) is a CAUcHY-sequence.

Concerning (4.14), without loss of generality, let f be real-valued. The function f, continuous on the compact maximal ideal space, is bounded by IIfllu Therefore, for any evaluation homomorphism h., If (h n)l Now put

-.g

'If" U.

n1= TTPk", n2= ITpk, P

P

where in the first Cresp. second] product the product runs over all powers pke for which f(pkp) is positive Cresp. negative]. Then f(n1) and If(n2)I are uniformly bounded and the theorem follows.

We use the following notation: given any arithmetical function, define (4.16)

f(P)(n) = f(gcd(n,p°D )), if p is prime,

and (4.17)

FR(n) = f( gcd(n,p>R n pOD))

The functions f(P) are fibre-constant.

Proof of Theorem 4.9. (a) Assume that (4.13) and (4.14) hold. The function f being additive, we obtain (4.18)

f = 2: PsR f (

)

+ FRS

Uniformly Almost-Periodic Arithmetical Functions

152

and the functions f(p) are in $u by Theorem 4.8. Next, IFR(n)I =

I

f(n) -P

R

f(p)(n)

if R is sufficiently large, and so f

c

I SpR sup If(pk)I < s

u

(b) If X = (0,0,...), 2" = (kp)p, where kp is arbitrary for p > R and k = 0 if p s R, then hx. is near ha, Since f is additive, we obtain if R is sufficiently f (ham,) = 0; f is continuous, and so If (hr. )I < .

E

large. Therefore, evaluating f (ham,.), one obtains I ZR
I

<E

for any system kp of exponents [kp= oo Is admissible, f(p°°) = limk f(pk)],

and so every subseries of Ep f(pkp) is convergent. Therefore this series is absolutely convergent for any choice of the exponents. This implies (4.14).

Proof of Theorem 4.10. (a) Assume that (4.13) and (4.1S) hold. Being multiplicative,

f = psR IT f(p,

FR,

where the fibre-constant functions f(p) are in ,$u. Next, using (4.15), 1-T f(p)(n) I s exp {pzR( I f(p)(n)I- 1) } S C,

uniformly in R, where * means that summation is only over those primes, for which f(p) (n)I i 1. And I

f(n) - TT psRf(p ) (n)I s

C

uniformly in n if R is large, again using (4.15). Therefore, f is in Su. (b)

If f is in $u and multiplicative, then the proof Is similar to the

corresponding proof of Theorem 4.9. The details, a little more compli-

cated than before, are omitted. It is helpful to use the fact that the absolute convergence of a product IT xiis equivalent with the absolute convergence of the series Z (x i -1). A second proof of the result for multiplicative functions is possible by reducing the assertion to the corresponding result for additive functions. First we prove the following lemma.

n/.4. Gelfand's Theory: Maximal Ideal Spaces

153

Lemma 4.12. Assume that f Is in ,$" and f(1) = 1. Then (1)

it

f(pk) = 0 Is possible for at most finitely many primes.

k m (2) For any s > 0 there Is an Integer R so that I f(n) -

II < s

If

gcd(n,R) = 1.

> S for some k is possible (3) If 8 > 0, the inequality If(pk) at most for finitely many primes. (4) If f is in £u and multiplicative, then If(pk)I S 2 Ilfllu for some k is possible for at most finitely many primes p. 1

I

.

1

proof. (1) If (1) is not true, then an ascending, Infinite sequence p1, p2, ... exists with the property f (h P, ,) = f pa) _ ... = 0. But the sequence of evaluation homomorphisms hp..' k =1, 2, ..., is convergent to h1 In AS. Therefore, 0=

k

lim

I (h P

,

) _ IE (h 1) = h 1(f) = f (l) = 1,

a contradiction.

(2) is proved using the same idea: if n is not divisible by the first r primes, r sufficiently large, then f (h n) is near f (h1) = I. (3) is obvious now.

(4) Let e > 0. For the evaluation homomorphism H1 the Integers j(r,H1) are ° 1 mod r. Choose R so large that for H R-near H1 [ this means

that j(r,H) = j(r,H1) for r = 1, 2, ..., R, no condition for r > RI If (H) - f (H1)I < E. Assume that n is coprime with R!; then there are integers x,y, so that I + R!-x,

and so f(n)-f(y) = f(1+R! x). The boundedness of f(y) implies If(n)I 2 Ilfllu 1

Choosing H =

then

.

I

Is near 1 and the result is proved.

In the proof of (4), complete multiplicativity was used. However, a variation of the proof also applies for the general case: Let e > 0. For H1, the evaluation homomorphism at 1, the Integers j(r, H1) are = 1 mod r.

Choose R so large, that for any H "R-near" to H1 [this means that j(r,H) = j(r, H1) for r = 1, 2, ..., R] If (H) - f (H1)I < 2. Assume gcd(p,R!) = 1; then, for every k in IN there are Integers x, y such that

Uniformly Almost-Periodic Arithmetical Functions

154

Pk

.y=1+

We may assume that p.}' y [otherwise take the solution x' = x + pk' y'

=

y

+

Then

R!].

2 If(i + R! x) I

IIfIILL-'.

f (pk) . f(y)

=

Choosing H =

f(1 + R! x), hence we obtain

If(Pk)1

If(1+R!.x) - 11 = If (H) - f (H1)I < 2. Therefore If(pk)1 > 2

for every prime p R!, and for every k E

IIfIILL-1

IN,

and the result is proved. Lemma 4.13. (1) Assume that f is multiplicative and in $u. If p Is fixed, ftt, a(pk) = a exists, then the multiplicative function and llmk

.

with values ftt(gk)

=

f(qk)

If q

is

a prime *

p,

and ftt(pk)

= a(pk) f(pk), is again in $u. (ii) With the same assumptions, the multiplicative function ftt with values ftt(gk ) = f(qk) if q is a prime $ p, and ftt(pk) = I for k = 1, 2,... , is in 2u. Proof. (i) is clear; f is multiplied by a fibre-constant function in ,$u. (ii) Choose F in 23, F R-even, a-near f, so that 11 f - F IILL < E. If prIIR,

then write R = R' pr, p4' R'. The function G: n H F(gcd(n, R')) is even; if n = pe.n', p4' n', then Ifa(n)-G(n)I = f(n')-F(n')I < s. Therefore G

is near fa, and so f isin,$u.

11

Now we give a second proof for one directionll in Theorem 4.10. Let f $u be multiplicative. We would like to look for g = log° f, but in E

order to do this some preparations are necessary. According to Lemma 4.12 the relation If(n) - 11 < i is true for all integers which are coprime

with some finite set 9 of exceptional primes. Change the function f into a multiplicative function f tt with values ftt (pk) = 1 at these finitely many exceptional primes. Then f is again in $u, for Lemma 4.13 (ii). tt Now the logarithm behaves nicely in the disc B(1, 2), and g = log ° f tt is additive, and again in $u by Corollary 2.5 (3). Then Theorem 4.9 shows E sup Ig(Pk)I 5 K, p

11 f e

k

.emu Implies the convergence of (4.1S). The other direction is simpler.

jy.5. Application of Tietze's Extension Theorem

and, using

s Ilog(1+z)I 5 Z IzI in

the inequalities

155

Izl s 2, this

implies

sup f(pk) S 2- K, E p not In t; k where p runs through non-exceptional primes. The finiteness of the (4.19)

1

I

I

other primes finally gives

Z sup p

k

I

f(pk) - 1

1

< oo.

IV.S. APPLICATION OF TIETZE'S EXTENSION THEOREM

Using our knowledge of A., AM and the TIETZE Extension Theorem [see, for example, HEWITT-STROMBERG (19651, or the Appendix, Theorem A.1.31, we prove the following theorem.

Theorem S.1. Given a sequence (nl) of (pairwise distinct) Integers greater than one with the property (S.1)

the minimal prime-divisors pmin(nl )= pl of nl tend to co as j -) co, and given complex numbers al converging to a E C, then there exists

a function f in 8u assuming the values al at n1.

.

hnl = hl in A. . The subset K = (hI) U (hnl) of A. is closed and therefore compact. Define a complex-valued function F on K by F(hi) = a , and F(hni) = al Proof. Condition (5.1) implies that liml--->

F is continuous on K, and TIETZE's Extension Theorem gives the existence of a continuous function FP on A. extending F, which Is the image of some f in .$u under the GELFAND transform, and n

f(ni) = f (hnl) = F(hnl) = al.

Uniformly Almost-Periodic Arithmetical Functions

156

Theorem 5.2. Given a strictly increasing sequence ne of positive Integers and given complex numbers ae with the properties lim ae = a exists, and (I) e

the evaluation maps hne E A" converge to some H in A.),

(ii)

then there exists a function f E Bu assuming the values ae at ne.

Proof. The subset K = {H} U (U (h ne}) is closed and therefore compact. Define a function F on K by F(H) = a, F(hn) = ae. Then F is e continuous on K, and by TIFTZE's Theorem F is extendable to a continuous function F* on AM. This function is, under the GELFAND transF* (hne) form, the image of some f E AM. Then f (ne) = hne(f) = = ae.

The definition of the topology of 7L^ immediately gives the following example.

Example. Given a strictly increasing sequence ne of non-negative integers with the property

given R E W, there exists an Z0 E W such that for every L z Z (S.2)

nL

ne

Z0

mod r for 1 s r 5 R,

then the evaluation homomorphisms hn?, are convergent and Theorem 5.2 is applicable. For example: (a) If ne = Z!, then condition (5.2) is obviously true. (b) If ne+1 = ne ue, and ue = 1 mod r for 1 s r s R(Z), R(Z) -> co, then the sequence hne is convergent.

IV.6. INTEGRATION OF UNIFORMLY ALMOST-EVEN FUNCTIONS

The GELFAND transform ^: u -4 e(A2), defined by ?(h) = h(f), is an isometric algebra-homomorphism. The inverse map is simply the restriction map

IV.6. Integration of Uniformly Almost-Even Functions

L: t°(A2) -) Bu ,

L(f*):

157

n H f*(hn), f*

is any where hn is the evaluation homomorphism at n, and where function in e(A2), the space of continuous functions on A$. Equation (6.1) is clear from L(f )(n) = If (h n) = hn(f) = f(n).

Examples. 1) Multiplicativity reads as follows: f ¢ Bu is multiplicative If and only if f (hnrn) = f (h n). f (hrn) if gcd(n,m) = 1. This result may be extended by continuity of f :Given H, H' in A$, represented by the vectors (k P ), resp. (kP' ), and assuming minkP, k P') = 0 for each prime p (so that H, H' are "coprime"), define the product H H' ')); then f , k as that homomorphism belonging to the vector (max(k ^ ^ P P ^ is multiplicative if and only if f (H H') = f (H) f (H' ) for all coprime homomorphisms H, H'. Similar remarks apply to additive functions. 2) We construct the image of the RAMANLUAN sum cP. under the GELFAND map ^. Let the homomorphism H in As be described by its vector of exponents {k }. Then put P if kP < k-1, 0, C(Pk,H) _ if kP = k-1, (6.2) cp(pk), if kP > k-1.

Clearly, this function C(pk,H), defined on arguments H in As, Is an And the extension of cP"the values C(pk,hn) being equal to k function H H C(p ,H) is continuous since the sets

d={HEAs k p ]} are open in A. So C(pk,

is the GELFAND transform of cP.. Using the multiplicativity of the RAMANUJAN sums with respect to the index, we obtain the transforms of all RAMANL[JAN sums cr. .

)

The mean-value M: Su -, C, f H M(f) is a non-negative (that is, f z 0 implies M(f) z 0 ) linear functional on Ru. Due to the obvious relation IM(f)I s Ilfllu it is continuous. The map (6.3)

M": F H M(L(F)), M": e(AB) -) C,

Uniformly Almost-Periodic Arithmetical Functions

158

is nothing more than an extension of the mean-value-functional M to L°(A2), and so Mu: t'(A ) -) C

a non-negative linear functional; it is continuous (I M"( F)I 5 IIFII ). Then Rtasz's Theorem (see Appendix A.3) immediately gives the following result. is

Theorem 6.1. There exists a complete and regular probability measure µ, defined on a o-algebra 4, containing the Bore] sets of h$ , with

the property fA F dµ = M"(F) = M(L(F)).

(6.4)

for every F E L°(02). So the mean-value M(f) = limX

can be represented as an integral,

_ x 1 nSX f(n) of functions f in

M(f) = fA f

(6.5)

In fact, it will be proved

h2 = 11 {

(6.6)

dµ.

that µ is a product measure. Write 1,

P = 11 IN,

p, pZ, ..., p°° }

and define probability measures µ

P

on the factors IN by P

µP(Pk) = p

(6.7)

1u

(iP(p

Then µP is defined on the Borel sets sets of IN ). The product measure

)

= 0.

of INP (

these are all sub-

P

(6.8)

11 µP

is defined on the least a-algebra f = 11 with the property that P P all the projections iP: A .i -* IN are f-78(N )-measurable ( this means P P that 7CP-1(AP)

E P for any Borel set AP in 78(N

P

Proposition 6.2. The product o-algebra JP = 11 P a-algebra of Borel sets in A A.

)

).

is equal to the

IV.6 Integration of Uniformly Almost-Even Functions

159

proof. Both the c-algebras mentioned in the proposition are generated by the measurable rectangles TT 7Z p, where 7Zp C NP and 7Z.p = NP for all but a finite number of primes p. This is true for .' by definition of the product (j-algebra; and by definition of the topology of A. it Is

clear that all the measurable rectangles IT 7.p are Borel sets, and that all these rectangles belong to the Borel sets. Example 3. Denote by 50 a finite set of primes, and, with each p e 9), associate an integer (including co) m(p), 0 s m(p) s oo. Characterize an element h in A. by the vector { kp(h)}p of "exponents". The set

Y = { h e 02: kp(h) = m(p) for each p in

9)

}

has measure p m(p).Zp

µ(Y) = fA XY dµ =

where tp = 1 - p-1, if m(p) < co, $p = I otherwise. The expression p-' is to be interpreted as zero.

Proof. (a) Let m(p) < co for each p in P. Y is open and closed, so the characteristic function XY is continuous, and µ(Y) =

lim

N-1

N -4c,

nsN m(p) In

The relation n e Y is equivalent to p iff gcd(a,b) = using Zdl (a,b)µ(d) = 1

m(pp )= mp and P = p `

1,

XY(n).

for each p in P. Therefore,

and writing P _

{

P1' "' ' pr}'

p r`r'', we obtain

,

21 N-' ' n5N X r Y(n) = d p µ(d) ...dp, µ(d) 1

N-1

1. N n=0 mod (Pd,...d,)

For N --) co, this expression tends to IT p-m(p).( I _ p-1 ). PET

(b) In the case where m(p) = co for at least one contained in every set

Zm={heAs,m(p)Zm} with measure

P in

the set Y is

Uniformly Almost-Periodic Arithmetical Functions

160

i1(Zm) = (1- p-1 ). (p-m + P-m-1 + ...) = p-m,

according to case (a), and thus µ(Y) = 0.

11

Example 4. The set N of positive integers (embedded in 02) N c A2 has measure zero. Enumerating the primes as p1 < p2 < ..., the measure of the set

Yr,s = { x ( As, mp(x) = 0 for pr S p s ps } is

r IT s

(

1 - pp

)

(according to example 3). Therefore,

Yr = (l Y r,s skr has measure IT ( I - p-1) = 0 and the assertion follows from IN c U Yr r

P>P,

coincides with the pro-

Theorem 6.3. The measure space (As, duct measure space

(TTNp,P,µ ). P

Proof. According to HEWITT-STROMBERG [196S] a product measure is

determined uniquely by the values of the measure on measurable rectangles. Without loss of generality, these may be taken as TT A p, where

AP = NP for p

z

po, and AP =

{pm(P)

;

m(p)

a

M(p)},

where

M(p) c {0} U N U {co}. Then

µ (TT Ap) = Y1p

IL (Ap) =

On the other hand, the same expression is obtained for µ(TT A ) P

example 3.

Corollary 6.4. If 9 Is a finite non-void set of primes, and f(p): NP -4 is µ P-Integrable for each prime p e 9', then the function f : A$ - C, h H TT f(P)(n (h)) pcT

P

is [L-measurable, and fo.Y3

f dµ

pTT C 71

fQom] dµp. O. P

Of course, iLp is the projection of 0$ to its "p-th factor" N .

by

p

C

IV.6. Integration of Uniformly Almost-Even Functions

161

Example S. The continuous extension of the RAMANWAN sums cr to "

AR was given mean-value is

(see (6.2)) as Cr

M(cr) = f

0R

=

.

x

H TTp"Ilr c' (r()P). Therefore, the

cr" dµ = IT fIN' c"P` dµ P P`IIr

Pk-1,(1-p 1).p-(k-1) + p(pk) .

-

pr

= 0 If r z 2, and l If r =

EmZk P

m'(1-p ') }

1.

Similarly, 2m(r).

7 (1-p), if r Z 2, pr

M(IcrI) =

if r =

1,

1,

and

f

mdilp = Pk-' (p _1) if k = m, otherwise = 0, P

and, therefore, the orthogonality relations M(cr'cs) = p(r) if r=s, and = 0 otherwise, are proved again. The final example 6 gives a calculation of the RAMANUJAN coefficients

for functions f in Bu which are finite products of fibre-constant functions f(P),

f = IT f(P), where 91 is a finite set of primes. pcT

equals f f cU dµ, and, this being a product

The mean-value over simpler integrals,

(1-p-1)( f(1) + p-1.f(P)+p 2f(p2)+...)

M (f'cr) _ TI

Pei),PXr

x TT

k PE9'.P II r

(1 - p-1).{-f(pk-1)+PP(Pk)'Emikp-'.f(pm)}

if all primes dividing r are in 91, and otherwise r

0.

An extension of the integral to the larger class (vector-space!) ,

q(

3 ( A2), µ), where q 2 1,

Uniformly Almost-Periodic Arithmetical Functions

162

of measurable functions F: 0B -4 C with the property fo., IFIq dµ < co is possible. Identifying functions F, G with IIF-GIIq :_ =

ffo.IF-GIq dp 11/q = 0,

the well-known L'-spaces are obtained. L2 is a complex HILBERT-space with inner product

F,G > = ro$

dµ.

The set of functions r = 1,2, ... is an orthonormal basis in L2. This follows from the fact that the continuous functions on A are dense in L2 and the linear combinations of [extensions of] RAMANUJAN sums are dense in L°(A2).

Finally, we note that a more powerful theory of integration of arithmetical functions was developed by E. V. NovosELov about 1962-1964, and the most powerful theory of integration, due to J.-L. MAUCLAIRE, is presented in his monograph [1986].

IV.7. EXERCISES

1) The pointwise product of an r-even and a t-even function is { l.c.m[r, t]}-

even. Prove this and a similar result for periodic functions.

2) Let r e N and f the indicator-function 1rN of the set r W. Calculate the RAMANLUAN coefficients a (f) and the FoURIER series of f. d

3) For given r e N, calculate the RAMANLUAN coefficients ad(f) for the function f defined by f(n) = if gcd(n,r) = 1 and f(n) = 0, if 1

gcd(n,r) >

1.

IV.8. Exercises

163

Solution: p (d)

` $

,

if dir,

e

a (f)

d

4)

Prove:

= 0,

if d Xr.

the quotient space £/$ is of infinite dimension.

(Hint: the residue-classes e1,r + .2, r = 3, 4, ... are pairwise different.) 5)

6)

The quotient space AID is of infinite dimension.

Let k be a positive Integer, and f an arithmetical function. Put fk(n)

= f(gcd(n,k)). Prove the equivalence of the following three

properties: (1) f e 2 ",

for every s > 0 there is a k in IN so that f-fk 11u - E. (3) the set { fk, k e N } is relatively compact in the set of bounded (2)

II

functions with the topology induced by II. Ii

.

7) Prove: the assumptions f e A, inf.,, If(n)I s 2, do not imply f-1 a A. (Hint: f(n) = 1 + 2 e,.(n).) 8) Let f e Au have no zeros. If If1-1 a A", prove that f-1 a '4u. 9) Give a formula for the GELFAND transform Cr a 0 NL(JAN sum cr a 2".

of the RAMA-

10) Describe a countable base for the system of neighbourhoods of the evaluation homomorphism h1 a A2.

11) Let {n1} be a sequence of positive integers with the property that the least prime-divisor pI of nl tends to infinity. Then the evaluation homomorphisms hn converge in A2 to h1. r

12) Let {n1) be an increasing sequence of positive integers with the fol-

lowing property: for every R e N there exists an io e N so that n) = nl mod r for 1 s r s R, j a I> i o. Then, in A.., the evaluation homomorphisms h n are convergent. [Example: nl = j!]

13) Prove: the evaluation homomorphisms hn, n = subset of A9).

1,

2, ...,

are a dense

Uniformly Almost-Periodic Arithmetical Functions

164

14) Show in detail that A. is homeomorphic to

= I {1, p, p2,

iN*

... ,

P,),

P

where each factor is the ALEXANDROFF-one- point-compactification

of the discrete space 0, p, p2,

... }.

1S) Let Xr s be the characteristic function of the residue-class s mod r. Mr

Prove: `Xr,s', otherwise.

l

Xr,s ) = 0 if (rt,r2)] Is1-s2I, and = {lcm[rt,r2]}

t

a

16) (A. HILDEBRAND).

a) Prove, for all Integers qt, q2, N, the asymptotic formula

N-t'Zn,N

q,(n)-cg1(n)

c

= Sq

b) There exists a positive constant ct such that the inequality N-t' 2:nsN I zgSQ a q' cq(n)12 s ct ZgSQ IagI2. cp(q) Is

true for all integers N, Q s N' and all complex sequences

(at, a2, ..., aQ).

c) Prove, by dualizing this inequality, qSQ


.

I f(N)*. (q)I2 s

for all Q s N' and complex values f (N) (d)

= N-t d

If(n)I2 f(1), f(2), ...,

f(N), where

n,N,n-O mod d f(n).

d) Prove, for all integers N and complex numbers f(1), f(2), ..., f(N) the inequality E

p-k I pk N-t.nsN

P` N

p" nf(n) -

N-t nZ N f(n)

I2 s c2 N-'-

n2N

If(n)12.

This is another dualized TuR&N-KuBILIUS inequality (see I, Thm. 4.3).

Hints: a) Use I, Theorem 3.1 (b). b) Estimate the error term with CAUCHY-SCHWARZ's inequality and apply WINTNER's Theorem II, Corollary 2.3. c) Use I, Theorem 4.2.

165

Chapter V

RAMANUJAN Expansions of Functions in $"

Abstract. This chapter gives the main parts of A. HILDEBRAND's dissertation, written In Freiburg (1984), which deals with the polntwise convergence of RAMANUJAN expansions f(n) = 1 Z<m ar' cr(n)

of arithmetical functions f . 2u, where the RAMANCUAN coefficients are the "natural ones" defined by

a,(f) = (9(r))-1'M(f'cr) 11u

is The assertion that the RAMANCUAN expansion for functions in convergent is shown to be equivalent to some other assertions, in particular to the following, which is to be proved: the 11.11 - norm of the kernel function SQ k(n) = E,, Q {y(r)j-' cr(k) cr(n) Is bounded. The proof uses estimates for Incomplete sums containing the MoBIUS function, for example for 1

M(n,z) = ZdIn,dsz µ(d), M1(n,z) = 2] dln dsz lt(d)' log(z/d).

Ramanujan Expansions of Uniformly Almost-Even Functions

166

V.I. INTRODUCTION

As mentioned already in IV.2, the RAMANLUTAN-FOURIER coefficients

of an arithmetical function f are defined as ar(f)

(1.1)

M( f Cr ), r = 1, 2, ...,

fi(r)

if the mean-values M( f Cr) occurring

in (1.1)

exist;

cr denotes the

RAMANUJAN sum

cr(n) =

dlgcd(r,n)

d

µ(

=

isasr, gcd(a,r)=L

exp {27Ci

r

n}.

In the case where the RAMANLUAN coefficients a r(f) exist, the RAMANUJAN expansion of f is the [formal] sum ar(f) Cr associated with f: (1.2)

do exist, and therefore For functions in .Y3" the mean-values the RAMANUJAN expansion (1.2) of functions f in . exists. Of course, general theorems on the pointwise convergence of arithmetical functions having a RAMANUJAN expansion (1.2) are highly desirable. We shall

return to this subject later (Chapter VIII). For bounded arithmetical functions there is some convergent series f(n) = 2: lsr< brr-cr (n), as was proved by J. SPILKER [1980]. A simpler proof, valid for every arithmetical function, was given by A. HILDEBRAND [1984].

Theorem 1.1. If f Is an arithmetical function, then there are complex

coefficients br, so that, for any n, (1.3)

f(n) = Glsr
We reproduce HILDEBRAND's proof. Put r* = 1 if r = 1, and r* = r V p Pr if r > 1; so r contains every prime factor to at least the second power if r > 1. The formulae for the values of the RAMANUJAN sum Cr' where

V

the index r is a prime power (see I, (3.4)) imply that ce(n) $ 0 is possible at most for indices r dividing n. So the sum b r Cr' (n) is a finite sum Ersn br We try to choose the coefficients br in such a manner that

V.1. Introduction

167

I rln,rsn

f(n) _

b

c ,(n)

r

r

for every positive integer. This is possible since the system (1.4) of linear equations can be solved recursively by b1, = f(1),

bn = {cn,(n)}-1 {f(n) - Erln,r Here,

I cn.(n)I = n $ 0,

and thus the system (1.4) is solvable.

Theorem 1.1 is not very interesting, because the coefficients br are not the "natural ones". Convergence of the RAMANUJAN expansion (1.2) for a large class of functions was proved by A. HILDEBRAND [1984].

Theorem 1.2. If f is an arithmetical function in

$u

then the RAMA-

NLUAN expansion a c (k) = f(k) r

15r
r

is pointwise convergent for k =

2, ...

1,

.

Closely related to this result are the following three theorems. Theorem 1.3. If f is an arbitrary arithmetical function, for which IIfII Is bounded and for which all the coefficients ar(f) exist, then for every k

sup

(1.6)

Q:1

IrsQ I

ar

r (k)

s c(k)

Ilfil u

I

Theorem 1.4. For any finite sequence br, r s Q0, of complex numbers the estimate max Z b r-Cr(k) QsQ.IrsQ

(1.7)

I

s c(k)

max n

I

I br- c r(n) rsQ.

Is true. We define the kernel function SQ k(n) by (1.8)

S Qk (n)

= ZrsQ

cr(k) cr(n).

Ramanujan Expansions of Uniformly Almost-Even Functions

168

Then the partial sums of the RAMANUJAN expansion may be expressed as rQ

ar(f)' cr(k) = M (f ' SQ k

).

Theorem I.S. There Is some constant c(k), depending on k, such that the estimate (1.10)

II SQ k

III

5 c(k)

holds for every Q z 1 , where I l f l l l = li sup up x 1 nZx If(n)I.

V.2. EQUIVALENCE OF THEOREMS 1.2, 1.3, 1.4, 1.5

It is not difficult to see that the theorems given in section V.1 (with the exception of Theorem 1.1) are equivalent, and It is not too difficult to see this. We prove this equivalence as follows:

5

3

4

4

is obvious, using the estimate I Z f(n)'g(n)I

The implication

nsx

5 11f Ilu' Ensx Ig(n)I: r Q ar(f)'cr(k)

M(f'cr)'{p(r)}_1

I

=

1,2:

'cr(k)

= IM(f'S Q,k)I 5

II

SQk

'

II

1

3

®. Put g(k) _

Ilf.u

br'cr(k); then the RAMANUJAN coefficients

ZrSQ.

of g are ar(g) = br, and so we obtain for Q 5 Qo I

XrsQ br cr(k)

according to

3

I

=

Xr-Q ar(g) ' cr(k) 15 c(k) ' Ilgllu,

I

, and

Ilgllu = max n

I

r Q br'cr(n) 0

I.

El

V.2. Equivalence of Theorems 1.2 to 1.5

169

. The sign-functions) fQk (n) = sign( S Q k(n)) is even modulo

(Q!); So, putting R = Q!, it has the expansion

fQ,k = ZrsR dr cr, where dr = ar( fQ k With the definition (1.8) of the kernel function S Qk we obtain II SQ,k III = M(I SQ kI) = M( f Q,k SQ,k )

M( fQ,k

_ 2: rsQ

Cr )

'rsQ ar( fQ k) c,(k), and, using

4

,

this is

s c(k) max

2

I

fQ k(n)

I

s c(k).

Given a function f in 2u, choose approximating functions fn we obtain for every k from B with the property IIf-fnllu -9 0. From

El =

.

5 c(k)

su p 12Q

Q

II

f-fn IIu.

If Q is sufficiently large, then lrSQ ar(fn)'cr(k)

GrSQ ar(f)'cr(k) = ErSQ

= ErsQ ar(f-fn)'cr(k) + fn(k), and thus If (k)

s

- E

rsQ

I

f(k) - fn(k)

I

+ I

rsQ

ar(f-fn)'cr(k)

5 (1 + c(k) )' II f-fnllu. 4

that r 4

This implication will be proved by contradiction. Assume is false, so that

3 k V c(k) 3 Qo 3 {br}

:

max I X QsQo rsQ

c(k)'max I Z n

1) slgn(x) = 1 If x > 0, slgn(x) =-1 If X < 0, sign(O) = O.

rsQo

brcr(n)I.

Ramanujan Expansions of Uniformly Almost-Even Functions

170

Therefore, we obtain the existence of integers Qn s Q.+ 1. s Qn+11 Qn /

Co'

integers Mn / oo, so that En>N {Mn}-1 s

QN-2 '

even functions fn = ErSQ partial sums at the point k,

satisfying IIfnIIu = 1, with "large"

I ErsW a sequence of even functions, {Mn}-1

FN = E nsN

fn.

this sequence is a II.IIu-CAUCHYsequence with limit F In Su. Then IIF - FN IIu s En,N s QN-2 Our goal is to show that the RAMANUJAN expansion of F is divergent For IIfnIIu = 1 and E

{Mn)-

<

co,

{Mn}-'

at the point k. The RAMANLLJAN coefficients of F = limN ar(F) = ar(F-FN) + ar(FN) = ar(F-FN) +

nsN

FN are

{Mn}-'-ar(fn),

and so, Isolating the single summand with n = N, we obtain J, Z

.

ar(F)'cr(k)

I

z

Ir Q

N

-n N {Mn} 2

ar(fN).cr(k)

{MN}-1

.

N

it QN ar(fn)cr(k)

fmN}-i.I

are large and that

Using the fact that partial sums ErSQ Icr(k)I s p(r) s r, this is 2 {MN} 2 {MN}

co, by

)2

- EN fm.)-, - (QN' )2 nN {Mn}

z (MN} - 0(1) as N

11F-FNII (QN'

nsN

I

QN

IIF-FN IIu EQH Icr(k)I.

En,N

EQw

- Ir

-4 oo,

choice of the integers MN.

O

V.3. Some Lemmata

171

V.3. SOME LEMMATA

In order to prove Theorem 1.5, claiming that the II.111-norm of the kernel-function SQ k is bounded [by a constant depending at most on k], some lemmas are necessary.

First of all, it is clear I see II, Theorem 3.11 that for a non-negative multiplicative function f, satisfying 0 s f(pk) s Yt,Y2k, where 0 < Y2 < 2,

the sum Z nsx f(n) can be estimated by (3.1)

21

nsx

f(n) s c1(Y1,Y2)

.

x.(log x)-1

.

exp( I p-1-f(p) ) psx

Lemma 3.1. Uniformly In x 2 1 and k E W the asymptotic formula

µ2(n)__ y(k) nsx,gcd(n,k)=1

k

y(n)

flogx+C+h(k)l+O( J

L

l

k) ) 1

holds, where i' Is EULER's constant, C = L° + 21 {p(p-1)}-1

log(p),

P

h(k) _

is strongly additive,

&

4(k) =d k

d_,

µ2(d) is multiplicative.

We remark that in H.-E. RiCHERT & H. HALBERSTAM 11974] the lower log x is given; this estimate is rather easily estimate Sk(x) 2 accessible.

Proof of Lemma 3.1. Put fk(n) = n ' cp(n)) if gcd(n,k) = 1, and fk(n) = 0 otherwise. Write fk = 1 * gk. From the Relationship Theorem (see Chapter III) or, simpler, directly from the values (p-1)-1

gk (p') =

-(p-1),p-1 -1, 0,

if m = 1, p k k, if m = 2, p k, if m = 1, plk, if m > 2 or (m = 2 and plk),

Ramanujan Expansions of Uniformly Almost-Even Functions

172

we obtain E n=1 fore,

n-1

Igk()I < w, and E n=1 n-1 gk(n) =

µ2(n)

Sk(x) = Ens,,gcd(n,k)=1

=E nsx

n-I.

k-1

y(k). There-

-1 fk(n) - E nsx n

cp(n)

gk(n)

Emsx/nrn-1

{log(x/n) + t? + O(n/x)l

= Ensx

= Ei + E2 + E3, say.

Turning now to the estimate E nsu

n = p(u),

{p(n)j-1

which follows from (3.1) (or directly by elementary considerations) and to the fact that gk(n) $ 0 is possible only if n = where nl is squarefree, n1Ik, gcd(n2n3,k) = 1, in which case the formula gk(n1n2n3) = i1(n1)'li(n3)

'

n3

'

{cp(n2)'(p(n3))-1,

holds, we obtain the estimate nZt

Igk(n

S )I

nk

µ2(n1)

n,k

<< t

nt Z/n,

iL2(n1)' n,

{w(n2)}-1

n,Y(t/__ {9(n3)}-1'n3

E/n {p(n2)} 1 (t/n1n2

(k)

t

a

nTk µ2(n1) ni

=

Partial summation (see 1.1) gives

E n-1 Igk(n)I << 4W-u_"

n>u

and thus E3

<<

x-1 .nsx E Igk(n)I << x-2

+(k),

and

gk(n) n

- k-i

W

(k ) +

C7

(x -5,

(k))

'V,3. Some Lemmata

173

Finally, using partial summation, g (n) x u-1 nsx

log(x/n) = f

kn

ft

1

1

gk(n)du

U-1- X n-1,gk(n) du +

log x - f

k-1.9(k)

g k(n) du n

log x - fl u

n_1 n- gk(n)

_

nsu

n>u

1

\

The integral from 1 to infinity is equal to n=1 n-1 gk(n) log(n), which can be evaluated in the usual manner, replacing log n by Zp.lln log(pk) and inverting the order of summation. This calculation is a little laborious and is left as Exercise 3. The result is

f1

n-t, gk(ndu

u-

n2 u

{

`Pkk'

P

{P(Plog(P) +

p k

p l log(p)}.

This formula concludes the proof of Lemma 3.1.

11

The proof of Theorem 1.5 rests on estimates of the following incomplete sums over the MOBIUS function: M(n,z) = Zdln,dsz µ(d) and

M1(n,z) _

(3.3)

din,dsz µ(d) log(z/d) = f u-1 M(n,u)du.

Lemma 3.2. Uniformly in z z

I

II n H M1(n,z) Ill << 1.

Lemma 3.3. Uniformly in z 2 x 2 1 lim

N ---> m

N-1

I

M(n,z)

I

log(2x)

( log(2x) \ 218 \ log(2z) )

where p1(n) Is the least prime factor of n, pnln(1) = co, and where (3.4)

S=I-

2)/log(2) = 0.0860713...

The more difficult result is the second one; Lemma 3.2 can be deduced from Lemma 3.3 in the following way. First, for pklln and u z 1, there

Ramanujan Expansions of Uniformly Almost-Even Functions

174

is an identity M(n,u) =

(3.S)

and so (3.6)

M1(n,z) =

f zz/ p

M(p

U-1

du.

For the proof of Lemma 3.2 we have to estimate the sum EnsN IM1(n,z)I.

We split this sum EnsN IM1(n,z)I according to the condition Pmtn(n) > z [resp. s z] and use (3.6) in the second sum ( with p = pntn(n), (Pmin(n))klln, n' = n/(PmIn(n))k N-1

E

n
IM (n,z)I s 1

N 1 nsN,pmu,(n)>z E + N-1 .

IM (n,z)I

E

nsN,p ,(n)sz

1

('z

J z/p

u-t

IM(n',u)I du.

Ordering according to p = pmin(n) s z , we obtain, after replacing n' by n, N-1

E IM (n,z)I s

nsN +

E

1

Ek

t

IM (n,z)I 1

p-k ,

zip

In the first sum, according to the condition pmin(n) > z and the definition of M1(n,z) there is only one divisor d of n with d s z, namely d = 1, so in this sum M1(n,z) = log z. The sum EnsN,p(n)>z I equals m. EnsN,gcd(n,k)=1 1, where k =P:Kz IT p, and this sum is (3.8)

EnsN,gcd(n,k)=1 1 = Edlk µ(d)

'

(d + e(d)) = N

2k l

+ R,

where Ie(d)I s 1, and IRI s t(k). So, for N - co, the first sum on the right- hand side of (3.7) approaches lim N --> m

N-1

E nsN,p_(n)>z

log z =psz rl (

1-p-1)

log z << 1.

Using Lemma 3.3 in the second sum, we obtain Lemma 3.2 after a short calculation.

The sum appearing In Lemma 3.3 is estimated by the CAUCHYSCHWARZ inequality:

V.4. Proof of Theorem 1.5

2

IM(n,z)I

Z nsN

175

s

E IM(n,z)12 nsN nsN, M(n,z)*O P.,.,(n)>x

Pm,,(n)>x

So, for the proof of Lemma 3.3 it is sufficient to deduce the following two lemmas:

Lemma 3.4. Uniformly In z Z x 2 lim

N --* ao

N-1

Z

1

IM(n,z)12

<< { log 2x }-1

nsN

p.,,(n)>x Lemma 3.5. Uniformly in z Z x Z

Nlim

N -1

Z

1

nsN, M(n,z)*O

1

« { log 2x }

1

log 2x log

2z

where 8 was defined in Lemma 3.3.

The proof of these two lemmas is given in section S.

V.4. PROOF OF THEOREM 1.5

In order to prove Theorem 1.5 (and thus the other theorems of this chapter) first we have to transform the sum defining the kernel SQ k(n), using cr(k) = Yk' Igcd(r,k) k' µ(r/k ' ) We obtain S

Qk(n) = Zk,

Ik

k'' Z

= Zk' Ik k'

2]

rsQ k' Ir

rsQ/k' (p(rk'

cr(n) [L(r/k' ) )}-1

Crk. (n) µ(r)

If r is squarefree, then factorize r = r' r", where r' Ik' and gcd(r",k') = 1, in order to obtain

Ramanujan Expansions of Uniformly Almost-Even Functions

176

SQ k(n) _

k'

k' Ik

','k' (n)'i1.(r') cp(r' k')

r' sQ/k'

r"sQ/k' r'

r'ik

p(r")

gcd(r",k')=1

Using the inequality Icr(n)I s p(r) and the abbreviation

k' (n) = 1rsz,gcd(r,k')=1 (cp(r))-1 where z > 0, the estimate (4.1)

TZ

µ(r)' Cr(n),

rXlk' ITQ/k' r' k, (n) I

s k Ik

k

s Ek' Ik

k

ISQ k(n) I

'

follows, and thus (4.2)

IISQ,k

II

I

Er' Ik'

IITQ/k,

r' ,k' III.

For (4.2) it suffices to show the estimate supy'I IITZ kill < ao

(4.3)

Replacing cr(n) by the usual sum over divisors of gcd(r,n) and inverting

the order of summation, we begin with

TZ k(n) = Edln d =

ErSz,dlr,gcd(r,k)=1

E

E

{p(d)}-1

r' sz/d

dln,dsz

S,µ(r)'i!(r/d)

{cp(r)}

{cp(r'

gcd(r',dk)=l

gcd(d,k)=1

The inner sum is known from Lemma 3.1. Inserting the result, we arrive at

(n) = w(k)

T

z,k

k

F(1) (n)

z,k

+

+

w(k) k

w(k) k

rC+h(k)) l J

F(2)(n) z,k

F(3)(n) z,k

(k) F(4)(n)), `

with the abbreviations FZ1k(n) = Edln,dsz,(d,k)=1 µ(d) Iog(z/d), FZ (2 )(n) =

Edln,dsz,(d,k)=l p(d),

FZ (3) (n) = Edln,dsz,(d,k)=1 µ(d) h(d),

and

z,k

1

V,4, proof of Theorem 1.5

177

FZ4k(n) = z-'. ZdIn,dsz,(d,k)=1

{p(d)}-1

Thus, in order to prove Theorem 1.5, it is sufficient to prove sup

II

F ('k

11

The treatment of the [semi-] norm N1

,

i = 1, 2, 3, 4.

II

F(4)111 is easy: we estimate z,k

< co

1

E IFZ k(n)I = N-'-z-'

{p(d)}-1

,

d3/2 4(d)

dsz

nsN

1

nsN/d

(d,k)=1

5

{p(d)}-1

.

Zd5z

.

d' 4(d).

N) gives

Partial summation (beginning with %sN the estimate 0(1) for the last expression, uniformly in z. {p(n)}-1

Next we show that, without loss of generality, one may assume that k = I.

for short, where Xk For the remainder of section 4, we write µk = is the characteristic function of the set of integers which are coprime with k. µk is multiplicative, and has a representation as a convolution µk = µ*hk, where hk is multiplicative, and hk(pm) = I if p1k, and zero Using this notaotherwise. The series X°°n=1 k (n) equals tion,

FZk(n) = Zdln,dsz Vk(d) - Idln,dsz d.,d"=d 21d..ln,d"sz hk(d") 2: d'I(n/d"),d' sz/d" µ(d') hk(d")

_

1(n/d"), Fz/d"

whence IIF(2)II z,k I s

dsz

k

(d)

II

F(2) z/d,I 1 11

s {p(k)j-1 k sup .a

F(2)111.

11

w,1

Similarly, a corresponding result is true when the upper index (2) replaced by (1). For the upper index (3), a careful calculation gives

FZ3k(n) = Zdln dsz,(d,k)=1 ii(d)

'

Zpld

P-1, log(p)

is

178

Ramanujan Expansions of Uniformly Almost-Even Functions

p-1.1og(P)

ZPln,psz,p1k 2:Pln,psz,pYk

'

2:dln,dsz,d=O(p),(d,k)=1 V(d)

1

log(P)

p

FZ/Plkp(n/P),

.

and a short calculation gives II

FZ3k II1 S

psz prk P-2'log(P)

Fz/p,kp II1

log(p)

k

5

II

9W

N?

P

(2)

IIFW

1

II1.

So, finally, the assertion of Theorem 1.5 is reduced to the problem of a uniform estimation for the following "Incomplete" sums over the MoBIus function: M(n,z) = dl dSZ

µ(d)

(2) = FZ (n) 11

and

M1(n,z) =

dI aSz µ(d)

log(z/d) = f 1 u-1 M(n,u)du [ = F(n) ].

Thus Theorem S follows from Lemma 3.2 and Lemma 3.3 (see section 3).

V.S. PROOF OF LEMMAS 3.4 AND 3.S

The proof of Theorem 1.5 will be finished as soon as we have proved Lemmas 3.4 and 3.5. For this purpose we need the following result on the MoBIUS function:

Uniformly in x z

1,

t 2 1, and d e IN the estimate

Enst,(n,d)=1 n i µ(n) << 4(d)

holds true, where

tt

min

1,

109 X

log2t

stands for the condition p.l (n) > x, and where

V.S, proof of Lemmas 3.4 and 3. S

179

L (1 4(d) = Edlk d-''µ2(d) = Ipk

+p

).

TI p, the absolute value of the sum Using the notation P(x) =psx p Y_

nst,(n,d)=1 n

1

il(n)

is equal to IE nst,(n,d)=1 s

n-1'µ(n)

'

Emlgcd(n,P(x)) µ(m)I m-1.11 2(m).Iznst/m,(n,dm)=1 n

mIP(x),mst

log-2(2u) in u >-I, which is a little Using the estimate Znsu stronger than the prime number theorem, the inner sum in (5.3), with slightly changed notation (u = t/m, dm = k), is equal to IEnsu,(n,k)=1

n-1µ(n)I

n-1. µ(n)

= IF1dsu,dlk°D d-1,

<< Ydsu,dlk° d-1

log-2(2u/d);

the notation dIk°° means: any prime divisor of d is a prime divisor of k. Splitting the sum X dsu,dlkm into Zds and Zd>-/-u, the last expression is d-1

<< log-2(2u)

+d"'

d'5'

d1

d/,ru) 3

<<

log-2(2u),

(k)

since

A

°O

(1+(p_l)-1)s2

d-1= plk IT

IT plk

(1+p-

).

and

d-'=IT dlk

plk

(

1

+

(p'-1)-

1

)s2

' nplk(1+p-f )

Therefore, IE nsu,(n.k)=1

n-1'µ(n)I << 4(k)' log-2(2u).

Inserting this result into (5.3) we obtain I1nst,(n,d)=1 n-1 µ(n)I << ZmIP(x),mst m 1µ2(m)(dm)/1og2(2t/m)

Ramanujan Expansions of Uniformly Almost-Even Functions

180

s 4'(d)

>mIP(x),m5t

By (3.1) 2:msu,mIP(x) (m) << u min(1, log(x)/log(2u)),

and partial summation immediately leads to assertion (5.1).

Now we come to the proof of Lemma 3.4. Uniformly in z z x 2 I we have to estimate lim

N-m

{M(n,z)}2.

E

N-1

(

The sum En,N,pm,,,(n)>x {M(n,z)}2 is equal to (remember that

tt

means

the minimal prime divisor of the variable[s] of summation is > x) lim N

a

- mN nsN,n-0 mod 1

lcm[d1,d2]}-1,

Ed d,sz

= V(x)

1

where V(x) = P:zX II ( I - p-1

) << ( log 2x )-i.

The argument needs an asymptotic evaluation of msM,p

(m)>x

which was given in (3.8). Using lcm[d1,d21)-1

{

=

{dId2}-1,zdld,,dl

d, 9(d),

we obtain

X"

d,sZ

lcm[d1,d21}-1

=

dSZ

2asz d-2

n-'-[L(n) 12 p(d)112(d)-(znsz/d,(n,d)=1n-1µ(n)12

The estimate of the inner sum was given at the beginning of this section. Inserting the result, we obtain E

tt tt

µ(d )'i1(d

lcm[d ,d

]}-1

<< Z

1Zx

tt

d

log (2z/d)

If z s x, then replace min(...) by 1. If z > x, then split the interval d s z into d s z/x and z/x < d s z. Replacing min(...) by 1 in the second sum

V.S. Proof of Lemmas 3.4 and 3.S

181

and by log2x/log 2(2z/d) In the first sum, we obtain

lcm[dl,d2])-1 << log2x . El +

Ed, d,,.

2,

where a

l =- dsz/x

d-2

u

212

log2(2z/d)

From (3.1), Ensu partial summation, E2 = z/xZ

#

1

E z/x
+ u/log(2x) if u z

d-29(d). 2(d) =

1,

z z/x

and so, by U-2. S(u)du << 1.

El contains an additional factor log2(2z/d) in the denominator; partial summation leads to E1 <<

{log22x}-1.

Thus Lemma 3.4 is proved.

11

A modification of a method due to ERDos and HALL is used to prove Lemma 3.5. Recall the notation M(n,z) = Edln,dsz µ(d). Then, In 2 < y < 2,

uniformly in 1 s x s z, with an 0 -constant depending at most on y, the relation (S.S)

lim N-1 I M(n,z)I . yn(n;x,y) N-,m nsN,x
<<

)2(y-1)

is true, where Q(n;x,z) =

x
is a completely additive function of

n.

The proof of (5.5) begins with IM(n,z)I = IEdsz,dln[L(d)I

Edin' ,z/p(n)
,

where n' = n/{p(n)}m, where p(n)mIIn and p(n) = pmin(n). This follows from the identity M(n,u) = M(n/pm,u) - M(n/pm,u/p).

Ramanujan Expansions of Uniformly Almost-Even Functions

182

We now obtain lim

N-

I )

nsN,x
N - ao

I

lim N -3 oa

= Zx
yn(n;x,Y)

.

dln',z/p(n)
(

N-1.

sN,p(n)=P,djn' Y

n(n;x,Y)

The limit in the last line is $ 0 at most if pm)n(d) > p. Using (3.1), in this case we obtain for this limit p-md-1,yr1(P'"d;x,z)

x

.

I

lim N-1

yn(n;x,Y)

nsN, p(n)>p

N->a,

m21

{ yn(d)/(Pd'(log p)Y)

<< Y

1

( log 2z

)Y-1.

Therefore, the sum to be estimated In (5.6) is << Y

( log 2z )Y-1

.

x
d-1.yn(d)

.

z/pY

Using, for u 2 1, the estimate Sp (u)

y n(d) <<

=

u

.

Y log(2u)

(I + log u log p

\

\Y

)

,

which follows from (3.1), we obtain for the inner sum, using partial summation, d-1,yn(d)

<<

(log(2z))Y-1

,

(logp)1-y.

Thus the left-hand side of (5.5) is <<

(log(2z))2(Y-1)

.

2:x
(p (logp

)zy-i)-i

Finally, partial summation shows that the last sum is <
This is

(

log(2x))2y-11-1

(5.5).

11

Proof of Lemma 3.5. The density to be estimated is limN-

1

ao

N

':nsN,P-(n)>x,M(n,z)*O I

V.S. Proof of Lemmas 3.4 and 3.S

183

= IimN where, using parameters x > summation are

1,

1

N-1 .1Z 1+ E2 1 + E311, 1

< y1 < 2,

z

< y2 <1, the conditions of

(1)

n s N, pmin(n) >z in X1,

(2)

n s N, x < pmin(n) :5z, (1(n;x,z) > x log( log (2z )/ log(2x)) in X2,

(3)

M(n,z) $ 0, x < pmin(n) s z, Q(n;x,z) s x log( log(2z)/ log(2x)) in 2: 3.

Obviously

lim

N -> m

= Ipsz ( 1-p-1 ). This is

N-'1

<< (log

s (log

2z)-1

log (2z)/ log(2x)) and Enlarge Z2 by replacing 1 by deleting the condition of summation for O(n;x,z). Then (3.1) leads to

Z2 «y (log <<

log

2z)-t

.

exp {

x
p-,.y 1 ( log(2z)/log(2x))-"logy, }

2x)-1

(log(2z)/Iog(2x))y,-1 -x-logy,

We replace the constant I in Z3 by n (n;x,z)-

I 5 Mn

log(2x))

2

and delete the condition of summation for O(n;x,z). Then (S.S) gives

13 «y. ( log 2x )-t ( log(2z)/log(2x))2(yz -1) Now we choose y1, y2 in such a way that the exponents of (log(2z)/log(2x))

become minimal, and then we choose x in an optimal manner: yt = x, y2 = 2x, x = (log

leads to an exponent (log

2)-1

2) - 1, and this is the assertion

of Lemma 3.5. Thus the proof of HILDEBRAND's Theorem is finished. 1) 1)

Some of the techniques used here appear In more refined form In HALL-TENENBAUM's proof of the ERDOS conjecture. All Integers

except a set of density zero have two divisors d, d' satisfying d < d' s 2d. See the monograph "Divisors" by HALL-TENENBAUM [1988].

Ramanujan Expansions of Almost-Even Functions

184

V.6. EXERCISES

1) According to Theorem 1.3, the partial sums of the RAMANWAN expansion

()

sup Q a

NI ZrsQ ar(f) cr(k) I < m

are bounded for every function f in Bu and every integer k. Prove Theorem 1.S by using (*) and the uniform boundedness principle (see, for example, HEWITT & STROMBERG [19651 (14.23) ).

2) The analogue of Theorem 1.S for uniformly-limit-periodic functions is wrong. More exactly: for every f E £u and every k E IN rsQ

21

lsasr,gcd(a,r)=1

f (a/r) e./r(k)

= M(f -

SQ,k),

where S

Q,k (n) =

rsQ

tsasr,gcd(a,r)=1 e-

a/r (n- k).

Prove that the sequence Q H II SQ k(n)II1 Is not bounded.

3) The function gk was defined at the beginning of the proof of Lemma 3.1. Evaluate the integral 1

u-1 , Zn'LL n-1 gk(n) du,

which occurred at the end of the proof of Lemma 3.1.

4) The estimates in Theorems 1.2, 1.3, 1.4 and 1.S are not uniform in k. Prove this (for Theorem 1.4) by the following example: if N E IN, p, then use P(N) = IT N
br(N) = µ(r) / (P (r), if r I P(N), and b, (N) = 0, if r4' P(N).

5) Construct an arithmetical function belonging to every space 2 , where q z 1, but whose RAMANUJAN expansion diverges for every integer n.

18S

NS-

A-,

Chapter VI

Almost-Periodic and Almost-Even Arithmetical Functions

Abstract. In this chapter, starting again with the spaces 2, £, and 4, a completion with respect to the semi-norm II f IIq = lim supX x-1 Znsx If(n)I q, q Z 1, gives the spaces $q, Z) q, and Aq of q-even, q-limitperiodic and q-almost-periodic arithmetical functions. Following J. KNOPFMACHER, It Is shown that these spaces are BANACH spaces. Next, the properties of these spaces are derived: functions In Al have mean-values, Ramanujan coefficients, Fourier-coefficients, f Al Implies IfI E A1. if f, g are in A and real-valued, then max(f,g), min(f,g) are also in A etc. The PARSEVAL equation Is given with two different proofs, and a result due to A. HILDEBRAND on the approximatibility of functions In 2 1 by partial sums of the RAMANUJAN expansion is given. Furthermore, a theory of integration Is sketched, and many arithmetical applications (mean-values, and the behaviour of power series with multiplicative coefficients) are given. E

186

Almost-Periodic and Almost Even Arithmetical Functions

VI.1. BESICOVICH NORM, SPACES OF ALMOST-PERIODIC FUNCTIONS

Chapter IV dealt with uniformly almost-periodic arithmetical functions; the BANACH-algebras considered there arose from the algebra

2 = Linc [cr, r = 1,

...]

2,

of linear combinations of RAMANWAN sums, respectively from the alge-

bra £ of periodic arithmetical functions,

D = Line [ea/r, r = 1,

1 s a s r, gcd(a,r) = 11,

2, ...,

where ea stands for the function e n H exp (2ni n), and from the algebra of linear combinations of the functions ea, E IR mod 7L, A = Linc [ ea,

(i

e

IR/7L].

These spaces were enlarged by the use of the supremum-norm Ilfllu = supncN I f(n)I,

(1.1)

and properties of the resulting spaces [in fact, these spaces turned out to be BANACH-algebras] (1.2)

u = 11. ll u closure(B), Bu =

11.

and A u =

11

.

II u closure(. ), II

-closure(A)

U

were given in Chapter IV. However, these algebras were rather small, for example the "smooth" arithmetical function n H n-SEp(n) is not in Bu (IV,Theorem 4.10). The "graph" of this function is given in Figure VI.1.

Of course, there are other norms on subspaces of the vector-space of arithmetical functions, and an enlargement similar to the one used above leads to other spaces which seem to be of greater importance in number theory. We use, for q 2 1, the BESICOVICH [semi-] norms (1.3)

Ilfllq

lim sup x t

={

,

1 f(n) Iq }l/q

2:

With the exception of the condition { f 9= 0 f = 0}, which is not true [ for example, e = v µ * 0, but llellq = 0 1, the properties of a II

1

11

yj.i. Besicovich Norm, Spaces of Almost-Periodic Functions

187

F i g u r e VI.1.

Values of the function n '- n-I. cp(n) in 1 s n s 598. The mean-valuefunction N '- N-1 ZnsN n -1 cp(n) is plotted In the same range. Its limit is 6 Z 7E

norm [listed in Chapter IV, section 1] hold true for II IIq, as is easily proved; for example, the proof of the triangle inequality [or MINKOWSKI'S inequality] uses (1.4)

{ lnlx I f(n) + g(n) Iq

}1/q 5 { En'x I f(n) Iq}1/q + { Ynsxl g(n) Iq}1/q.

This follows from HOLDER'S inequality Y-ne

,. an - bn) 5 { Zne 9- I an Iq

}1/q

{

Iq' } na S' Ibn

1/q.

for sums. The summation runs over a [finite] set S'; q is restricted to 1 < q < co, and q-1 + q'-1 = 1. An enlargement of the spaces B, 9, A by forming closures with respect to II f II as mentioned above, gives q,

- the space of q-almost-even arithmetical functions, Bq

=

II. IIq - closure of 2,

- the space of q-limit-periodic arithmetical functions, Z)`i =

II.

IIq - closure of 9,

Almost-Periodic and Almost-Even Arithmetical Functions

188

- and the space of q-almost-periodic arithmetical functions, Aq = II.

II

q

-closure of A.

So, for a function f in, say, $q, and for any

0, there exists a

>

E

[finite] linear combination t of RAMANL[IAN SUMS E-close to f with respect to the semi-norm 11.11 q, thus II f - t IIq < E. In the sequel, we will

often speak, inaccurately, of "the norm" correct term "semi-norm" 11.11

instead of using the

11.IIq

q

It is clear that ,$q, Dq, and .94q are C-vector-spaces; heuristically, one

is inclined to expect that properties of .$, £ and A are also valid in .1q, Dq and 94q, and this principle often turns out to be successful. But these spaces are not algebras [so the heuristic principle just mentioned is sometimes not applicable], neither with convolution nor with the pointwise product. For example, 1 ,1, but t = * is not in .941 because functions in Al have a mean-value (as will shortly be established in Theorem 1.2); does not have a mean-value. The function E

1

1

c

f(n) = (log p }2/3 if n = p is a prime, and zero otherwise, is in 941. Due to the scarcity of the primes (because of weak versions of the x-1 prime number theorem the result limX Tt(x) (log x)2/3 = 0 is true) this function is arbitrarily near to zero with respect to II. II1, but the [pointwise] square does not possess a mean-value and so f2 is not in A1.

The [obvious] inclusion relations ,$ c 2 c 4 imply

$gcZ)gcs4q,where q>_ 1. For r < q, HOLDER'S inequality gives 2:nsN If(n)Ir 5

{XnsN

If(n)II}r/q

:n-N

therefore, (1.5)

II

f IIr 5

II

f II

q

1f r s q,

and so 2q c ,8r, Dg C Z)r, and .94g c Ar

,

if r s q.

}q/(q-r)

W.I. Besicovich Norm, Spaces of Almost-Periodic Functions

189

Thus we obtain Figure VI.2, showing the inclusion relations between the various spaces defined up to now.

2)1

r

r

$1

1

Inclusion relations between spaces of arithmetical functions, for

\

Cl

$r

Dq

\

u

s r s q< m.

\ F i g u r e VI.2.

A general, simple, but none the less useful principle is given as the following lemma.

Lemma I.I. Suppose {An} is a sequence of linear functionals from a complex vector-space X (with semi-norm II . II) Into C. If for every xEX lim sup IAn(x)I s c - IIxii,

(1.6.1)

n-a

and

(1.6.2)

r

j`

there is a dense subset E of X such that the sequence

{An(x)}n=1,2,.., is convergent for every x in E,

then

(1.7.1)

the sequence (An (x)}n=1,2,... converges for every x In X,

(1.7.2)

the map A: x H

lim

n - co

An(x) is a continuous linear

functional on X, and IIAII s c.

Remark. This lemma and its proof are modelled after Exercise 18 in W. RUDIN 11966], p. M.

Almost-Periodic and Almost Even Arithmetical Functions

190

Proof. (1) Let E > 0, x e X, and let {xm}m=12 elements of E converging to x. Put c

m

=

lim

be a sequence of

n m ),form=1,2,.... _ A(x

There exist integers m, no such that for every n 2 no c- IIx- xrnll < a E, IA (X-X )I S c

E, and IA (x )-c

IIx-xmII +

I

<

y

E.

Then, for every k, n > no we obtain IAn(x) - Ak(x)I s IAn(x-xrn)I + IAn(x=n)-crnI + IAk(x-xrn)I + IAk(xrn)-crnl < E.

This proves (1.7.1). Then (1.7.2) is obvious from (1.7.1) and (1.6.1).

11

Theorem 1.2. Assume that f is an arithmetical function In A1. Then the mean- value M(f) =

(1.8.1)

x

lim

x-1

nsx

f(n),

the FoURIER-coefficients

f (a) = M(

(1.8.2)

a real,

and the R4MANujAN (or RAMANUJAN-FouRIER) coefficients ar(f) =

(1.8.3)

r = 1,

2, ...,

{cp(r)}-1,M(

exist.

Proof. For the mean-value, consider the maps AN: AI -3 C, AN (f) = N-1

.

nsN

f(n).

These maps are linear and lim supN IAN(f)I S Of III. The mean-value M(f) = limN AN(f) exists for every function in A. This vector-space is II.II1-dense in A 1; therefore Lemma 1.1 gives the existence of M(f) for every function f in A 1. The assertions for FouRIER and RAMANUJAN coefficients follow from the fact that and are in A 1. Another application of Lemma 1.1 Is provided by the following theorem.

VI.1 Besicovich Norm, Spaces of Almost-Periodic Functions

19t

Z)q, where Theorem 1.3. Let f be an arithmetical function in

q z 1, and g Is another arithmetical function. If g is bounded, In the case where q = I [respectively If Ilgllq. < oo In the case where q > 1J, q'-1 + q -t = I (as usual), and if where lim

x-1

x -->

21 nsx, n®a mod r

g(n)

exists for every pair a, r of Integers, then the mean-value lim

(1.10)

X

x-1

m

Z

nsx

exists.

proof. We apply Lemma 1.1 using the maps AN defined on 2q by AN (f) =

nsN

f(n)'g(n)

HOLDER'S inequality gives II g Ilu,

lim sup N--> m

I AN(f)l s c

1 1f1 1

,

where c =

if q = 1,

II g IIq if q > 1.

The value of AN at eb/r is

AN(eb/r

Osa
(a).N1

(eb/r

nsN, n=a mod r

Using (1.9), we see that the sequence {AN(f)}N-12

every function f in 2). Hence, the existence of

g(n)).

is convergent for follows from

Lemma 1.1.

Remark. This proof does not seem to work for functions f e

,P4

Examples.

(1) For an integer r > 0 let g be the indicator-function of the set r W. Then, for every function f e 2 1, the "mean-value with divisor-condition" Mr(f) =

lira X

m

X-1

nsx,rln f(n)

exists. The same argument shows that -1

xu rn co

x

:nsx,n = s mod r f(n)

exists for every residue-class s mod r.

Almost-Periodic and Almost Even Arithmetical Functions

192

(2) For the function g = µ2, the square of the MOBIUS function, assumption (1.9) is easily checked by [well-known] elementary calculations, relying on the convolution- representation µ2(n) = Ed-In µ(d) (see II § 2).1) Therefore, according to Theorem 1.3, the "mean-value on squarefree numbers" x

lim

x-1 eo

'

n

sx, nsquarefree sfree f(n) = lim- m

exists for functions f E .

x-1. Z

nsx

µ2(n) f(n)

1

(3) For g = µ assumption (1.9), in the case where gcd(a,r) = 1, is guaranteed by properties of DIRICHLET's L-functions. The case where gcd(a,r) = d > 1 can be reduced to the case mentioned above as follows: if d = p1 p2 ... ' Pk with distinct primes, and a =a'- d, r = r'- d, then x

lim

x-1 oo

z

nsx,nma mod r

µ(n) = (-1)k

d-1

lim y 1

y-3m

msy,m =a mod r µ(m)

exists. So (1.9) is valid for g = µ, and, according to Theorem 1.3, the mean-value lim x -3 m

exists for every function f E

X-'-nsx X µ(n) f(n)

D1

The spaces X39, D9, and 49 are complete: any 11.11 q-CAUCHY-sequence of functions in, say .89, has a limit, which is again in $9. Thus, following J. KNOPFMACHER [1976], we prove the completeness of these spaces.

Theorem 1.4 [J. KNOPFMACHER]. For q z .Z)9, and .s49 are complete.

1,

the normed spaces 29,

Proof. The spaces B9, Z)9, and 3 q are closed subsets of the vector-space V9 = {

f: N -C; Ilfllq < w ).

So it suffices to prove the completeness of V9, where q z 1.

1)

Another possibility for proving this Is: the function g = µ 2 is in 8 1 and so assumption (1.9) Is valid by example 1.

VL1,

Besicovich Norm, Spaces of Almost-Periodic Functions

193

Let {fk}k be a CAUCHY sequence in Vq. There exists a sequence {Ek}k

of real positive numbers, converging to zero, with the property II

ft - fk IIqq< Ek for every $ > k.

We are going to construct a sequence {xm}m of non-negative numbers with the properties (a) 0 = x0 < x1 < x2 < ..., xm --j

CO'

(b) the function f, defined by f(n) _

ft(n), if xR-1 < n s xQ,

satisfies

x1Y - nsx I f(n) - fk(n) I`1 < 2 Ek for every k in N, and for every x Z xk.

These properties imply I I f - fk 11q s 2 Ek for every k; therefore

so f E Vq and f give the

is

I I f1 1

q < co,

the limit of the sequence fk. So it remains to

[inductive] construction of a sequence {x m} m=1,2,. . with

properties (1.11).

For any integer k, 0 < k < t, there is a real number xk2 > 0 such that (1.12)

x-1 Ensx I fe(n) - fk(n) Iq < Ek' for every x z xkR.

Put xo = 0, x1 = max {x12, 1). Then x-1

(A12)

nsx I f2(n) - fI(n) Iq < E1, for every x z x 1,

x12xo +

(D1)

1.

We assume now that 0 < x1 < x2 < ... < xm are chosen with the properties (AkE)

k < 9 s m+l:

(Bke)

k < E s m:

(C) k

ksm:

(Dk)

k s m:

nsx I f1;(n) - fk(n) Iq < Ek, for every x 2 x1,- 11

x1

I ft(n) - fk(n) Iq < Ek (xC xt-1 ),

Z xE-1<nsx,e

F,

2< k xk

Z xg-1<nsxk

xk-i +

I fC(n) - f k(n) Iq < E k . x k ,

1.

Given any k, 0 < k < m+l, there exists a positive Ek < Ek, such that

Almost-Periodic and Almost Even Arithmetical Functions

194

x-1'nsx E Ifm+1 (n) - f k(n) Iq < £ k for all x k xk ,m+1 '

,

If Xk Z max { xm (1 - £k"k)-1' Xk m+1 }1 then

(n)-fk (n)I' E xm <nsxk' If m+l

s

nE

If m+1

2c X k

(n)-f k (n)I`'

< £k ' Xk 5 £

(xk - Xm).

Now take xm+1 so large that for k < m + I (Bk,m+1)

Exm<nsxm+1

Ifm+1(n) - fk(n) I4 <

£k

' (xm+1 - Xm)

holds, and for k < m+2: (Ak ,m+2)

x-1 -E

Ifm+2 (n) - fk(n) I4 < £k' V x 2 Xrn+1'

E a<m+1 E XP_l<nsxu

(C m+1 )

Ifa (n)-fm+1 (n) Iq < £ m+1

Xrn+1 Z Xm +

(Dm+1)

x

m+1'

1.

Thus, we obtain a sequence {x,,,} with properties (a), and (Ak e), (Bk,e),

(Ck) for all k < Q. The function f, defined as in (b), has the desired property V k e IN, V x s xk, and m e IN defined by xm s x< x m+1 E nsx

If(n)-fk(n)I4 = Ee
k<esm Ee-1<nsxe < £ k Xk + £k

fe(n)-fk(n) Iq I

Ife(n)-fk(n)I9

Ek<esm ( x

x

1 1

Ifm+1(n)-fk(n)I'

+ Exm<nsx

)+£ k'xs2£ k )

This proves the completeness of V9.

X.

11

The null-spaces are defined as follows: jy(Aq) = { f

e

.419,

II f IIq = 0

and similarly ff(29) and A'(29); these are subspaces of 349, resp. Dq,

W.I. Besicovich Norm, Spaces of Almost-Periodic Functions

19S

reSp, $q. The null-spaces are closed; the limit f of a II . 11q - convergent sequence of functions in, say, HY(Aq), is in 44q and has norm II f Ilq = 0. We denote the quotient spaces .A4q/JY(Aq), etc., by

Aq = 4q/JY(Aq), Dq = Dq/JY(2q), and Bq = q/jy(Sq). There

is a canonical quotient map it, n = nAg: .A4q _> '4 q/JY(Aq), IT Ac(f) = f + lY(Aq).

The quotient norm is defined by (see, for example, RuDIN [1966], 18.15) IIn(f)IIq = inf {

II

f+g

11

q,

g E Jr } =

II f Ilq.

Then Aq, Dq, Bq are BANACH-spaces.

In Chapter IV, Theorem 2.9, a uniqueness theorem was proved for functions in Du. In ,A41, a theorem of this kind is not true. However, arithmetical properties such as additivity or multiplicativity have consequences on the uniqueness of functions In A1. As examples, we prove the following theorems. Theorem I.S. (Uniqueness theorem for additive functions). Assume that

f and g are additive functions In A1. If

II f - g II1

= 0, then f = g

identically.

Theorem 1.6. (Uniqueness theorem for multiplicative functions). Assume f and g are multiplicative, and both are in £1, Ilflll * 0, Ek>1 p-k If(pk)l < m for every prime p, and f-g is in the nullspace JY(41). Then f = g identically.

Remark 1. The assumption IIfII1 * 0 in Theorem 1.6 is necessary. The functions f = E, g(n) = 1 if n = 2k for some k, and g(n) = 0 otherwise, are both multiplicative and satisfy IIf - gill = 0, but f * g. Remark 2. The finiteness of the norm Ilfll for some q > 1 implies (see g III, Lemma 5.2) Ekk1 p k f(k)1 < oo for every prime p. So this condition can be omitted In Theorem 1.6 If f E Dg for some q > 1 is assu-

med. We shall see later (VII, Theorem 5.1) that this condition is also superfluous in the case q = 1.

Almost-Periodic and Almost-Even Arithmetical Function,

196

Remark 3. If f E D1 is multiplicative and non-negative, then the condition MP(f) < M(f) implies Ik21 P k. I f(pk) I < co (see Exercise 4 ).

Proof of Theorem I.S. Put h = f - g, and let pk be a fixed prime power. For any Integer N, we obtain the lower estimate ZnsN Ih(n)I

2:nsN,P`Iln Ih(n)I = JmsN/P`,Pkm Ih(pk) + h(m)I

2t

h( Pk)I

2

- G msN/p`,p4m ' 1-

msN/p Ih(m)I.

Dividing by N, for N -) co, the inequality IIhIII 2 Ih(Pk)I

.

. (1_p-1) _ p-k.11h1I1

P-k

is obtained, and the assumption IIhJi1 = 0 implies h(pk) = 0.

11

Proof of Theorem 1.6. Assume that there is an integer n0 for which f(n0) $ g(n0 ); then there

is

a prime-power pk with f(pk) $ g(pk)

Then, for every N, Z If(n)-g(n)I 2 nsN msN/p1,gcd(m,p)=1

I f(pk) - g(pk)I. E

I

I

If(m)I - Ig(pk)I

msN/p` p .I' m

.

E If(m)-g(m)I. msN/p`

Add the term I

f(pk) - g(Pk)I

.

msN/p°,PI*n

If(m)I,

to both sides of this inequality and divide the resulting inequality by N ( pk (N/pk) ). Letting N tend to infinity, we obtain, using the abbreviation MP (g) = lim"-> - x 1. 1nsx,p1n g(n) [for the existence of this mean-value, see Example 1 following Theorem 1.3], II

f -gill + If(pk) - g(Pk)I P-k .

MP(Ifi)

z If(pk) - g(Pk)I P-k

M(Ifi) - Ig(pk)I p-k

II

f_ g III,

and therefore M (Ifi) Z M(If)I, a strange result certainly, which comes P from the assumption f $ g, which is to be refuted. Next

VI.2.

properties of q-Almost-Periodic Functions

GnsN,Pln

197

If(n)I = 21 k21 XnsN,P"iIn If(n)I,

= Zkz11f(Pk)I ( msN/p" If(m)I - ZmsN/P`,PIm If(m)I), and so

Zk2O If(Pk)I "msN/p`,pIm If(m)I = Ek21If(pk)I

' Z n N/p If(m)I

Dividing by N and using the dominated convergence theorem (this is p_k If(pk)1 < oo ), we obtain possible for y = 1 k21 (1+Y) MP(IfI) = Y' M(IfI),

therefore, using the estimate MP(IfI) z M(IfI), proved above, Y (1+Y)-1 - MOM ;, MOM,

which contradicts the assumption 11f111 = M(Ifl) > 0.

VI.2. SOME PROPERTIES OF SPACES OF q-ALMOST-PERIODIC FUNCTIONS

As mentioned already in section 1, HOLDER'S inequality II1 s IIfIIq

II

IIgIIq

, ,

where q-1+q'- I = 1,

implies .v4q c s4r c '41 whenever 1 s r s q land there are corresponding

results for the other spaces - see Figure VI.2]. Starting with k = 2 (which is HOLDER'S inequality), mathematical induction gives the following

Proposition 2.1. Assume that

1/q1+l/q2+...+1/qk=1,

(2.1)

where 1 < qx < oo. Then (2.2)

11

f1 ' ...

fk 111 s IIf1IIq

IIfk11q.,

Almost-Periodic and Almost Even Arithmetical Functions

198

Proposition 2.2. Assume that all the norms appearing In equation (2.3) below are finite. Then the following assertions are true:

(2.3)

(i)

If r t

(ii)

If

(iii)

II

I

1,

q-1 + q'-' = 1, then

s q s r, then IIq 5

II g

11U.

f II,q

II

II f III 5 II f 11q 5 II f IIr 5

II g 11'.q

, .

II f IIu

II f IIq.

Proof. (iii) follows from the definition of II.IIq; the other inequalities are obtained from HOLDER'S inequality.

Theorem 2.3. Assume that 1 5 q s r < co, and q-1+ q'_ I = 1. Then (1)

2u c .fir c 8q c 21, D C Du C Dr c Z) q c Z A c v4u c Ar c Aq c A1. c

.$

(2)

11q c Z)q C Aq C A1.

(3)

2u.Bq c $g,

(4)

2q- 2q

(5)

If f

E

if f if f

E

c

..

c 21,

g c Dq, D

,$g, then Re(f), Im(f) and IfI y) q, then Re(f), Im(f) and IfI 34 q, then Re(f), Im(f) and IfI

34u.Aq c 34g.

Aq.,4g E

.Sq,

E

Dq,

E

c 341.

34q.

(6) If f, g are real-valued and both are in ,$g [resp. Dq, resp. 3487, then

max(f,g) and min(f,g) are In Bq [resp. Z) q, resp. 8q].

Proof. Assertions (1) and (2) are clear. For (3), assume that f 34g, g E Au2 E > 0; choose functions G, F in A near g, f such that E

II g-G IIu < E/(II f IIq+1), II f - F IIq < E/(IIG IIu+1). Then F- G is in A and 5

IIq + IIG'(f-F)Ilq 5 IIg-GIIu'II f IIq +

f-Fllq < 2E.

34c1, g E 4g , E > 0, choose F, G E A, II f-F IIq < E/(II g IIq+I), 1Ig-GIIq. < E/(II F IIq+1). Then F G Is in A and (using HOLDER'S inequality)

(4) If f

11

E

5

11

1+

5 11 f-FIIq'IIgIIq, + 11FIIq'IIg-GIIq. < 2E.

V1.2. properties of q-Almost-Periodic Functions

199

(5) The real or imaginary part of a function in 2 [resp. 21, resp. A ] is again in B [resp. 2), resp. A ]. If F is in S or 2, then IFI is even or periodic and so, again, is in B or D. And, using the usual approximation s f-g 11 1, the assertions arguments [and the inequality Ifl - IgI Aq Implies Ifi in 41 q. But in are proved with the exception that f in if F in A, the WEIERSTRASS Approximation Theorem gives: this case II

then IFl in

Au

II

II

(by IV. Theorem 2.2), and this is sufficient for a proof

of the remaining assertion.

Assertion (6) follows from the formulae max(f,g) = 2(f+g) + 2']f-gl, min(f,g) = 2(f+g) -elf-gl.

11

Theorem 2.4.

(1) If f Is in A1, then the mean-value M(f), the FouRIER coefficients and the RAMANUJAN coefficients NO = ar(f) = {9(r))-'- M( f ' cr )

(2.S)

exist.

(2) In A2 (and so In the subspaces .82, 2)2) there is an inner product < f,g > = M( f-9_)'

(2.6)

and the CAUCHY-SCHWARZ Inequality (2.7)

I

15 11f112'IIg112

holds.

(3) If f is in

A2,

then at most denumerably many FouRIER coefficients are non-zero, and BESSEL's Inequality EpCR/ZIf (a)12 s Ilfll22

(2.8.1)

holds. If f Is In 22, then BESSEL's inequality reads (2.8.2)

211sr<_ w(r)

(4) The maps from

.v41

Iar(f)12 s Ilfll22.

to C, defined by

f H M(f), f H P(0), f -)a r(f)'

are linear and continuous. Moreover, the first is non-negative.

Almost-Periodic and Almost Even Arithmetical Functions

200

If (a)I s M(Ifl), lar(f)I s M(Ifl)

(5)

Remark. Some of the assertions of this theorem have already been shown to be true in section by applying a general principle from functional analysis. In spite of this, we give an ad-hoc proof again 1

here.

Proof. (4) and (5) are obvious. For (1), the existence of the mean-value has already been proved (in section 1); a simple, direct proof for (1) is to be given in Exercise 1. The functions and are in Al again, and so the FOURIER coefficients, which are mean-values, do exist. is In A1, and so its mean-value, which is the Inner

(2) The function

product, exists. The usual properties of an inner product are easily verified (note: < f, f >= 0 implies II f 112 = 0, but not necessarily f = 0). The method of proving the CAUCHY-SCHWARZ Inequality is standard in linear algebra. The same Is true for BESSEL's Inequality.

(3) The functions e0' , a c IR/7L, are an orthonormal system. Using only finitely many FOURIER coefficients f (a), we obtain

0 s< f - I f

f-Z

= - E f

I

=

-

'(13)e0

f (a)'<e«,f > + 1

F(a) ?(P)

ep>

F (a)12. I

Corollary 2.5. Assume that q > 1, and k z 1 Is an Integer. [resp. ,q ], then the "shifted" function (1) If f Is In A q n H f (a+n) is In Aq [ resp. £ q], where a is in Z. 1) f (a) : (2) If f is in A q [ resp. D q ] then, for b E W, the multiplicatively shifted function fib ) : n H is in Aq [resp. £ q ]. (3) If fl fk are in Ak, then the function F Is in A1, where k

(2.9)

F(n) = IT fx(bxn+ax), bx E W, a xE 7L. X=1

Proof. (1) Given E > 0 there is a function F in A, F = a e near f, Ilf-Fllq < I. Then Fiai(n) _ (a,,, ea(a)) e (n), and so F(a) Is in A and 1)

We assume that f(a+n) =

1

as long as a+n s O.

VT.2.

properties of q-Almost-Periodic Functions

201

IIf(a)-F(a)liq = 11f-F11 q < E.

then G is in A [resp. £7, and

(2) Choose F as above, put G: n H

x-t'EnSx

G(n) Iq s

If(m)-F( m)Iq,

and we obtain Ilf(b )_GIIq < bt/q E. (3) The case where k = 2 is obvious from (1), (2) and Theorem 2.3. The case where k > 2 is left as an exercise. Theorem 2.6.

(i) If g Is In At real-valued and bounded, then, for any e > 0 there exists a function t in .v4" near g, g - t IIt < s, with the addiII

tional property IItIIu S IIgIILL

(ii) If g E Al Is bounded, then g E Aq for every q 2 1. (iii) Assume that g e Al has a bounded representative and f is in Aq. Then the pointwise product is in a4q. Remark. The same results are true for the other spaces 2q and 2q. Of course, in (i), in these cases t may be taken to be in $ [resp. in D]. If g is complex-valued, in (i) it is possible to find a t satisfying IItIIu s

IIRegII2 + IIImgIILL

s 2 IIgIILL.

Corollary 2.7. If f Is In Dq, then the functions where x Is a DIR7CHLET character, Ik is the characteristic function of the set of Integers relatively prime to k or the characteristic function

of the set of integers congruent to a mod k, where gcd(a,k) = and the pointwise product

1,

are In Dq again.

Proof of Corollary 2.7. The functions x and 1k are periodic and bounded.

The function µ2 is bounded and is in $t (this is a consequence of the Relationship Theorem from Chapter III; it2 is bounded, therefore in ;, and it is related to the constant function I y°*). E

Proof of Theorem 2.6. (1) Given E > 0, choose a real-valued trigonometric polynomial t In s4 [resp. D or 21] near f, II g-t < E. Put W

w

II

t = max{ min (ta, IIgIILL), - IIgIILL}. Then IIg-tilt 5 IIg-t*Iit < E, and t is I. v4u

[resp. 9, resp. $ for the other spaces], and IItIIu S IIgIILL (ii)

is a special case of (iii).

Almost-Periodic and Almost Even Arithmetical Functions

202

0, choose t1, t2 in Au, such that Ilf-tlllq < E, IIg-t2111 112 < Eq/(1+llgllu2+ Iltl ), and lit2llu s 2- llgllu. Then an easy computation (iii) Let E

>

shows it

g-t2 IIq s { 4q-1 Ilglluq-1

IIg-t2111 }1'q.

Therefore, Ilfg-tIt2llq 5

S

S llgllu.E +

llgllu ll(f-tl)llq +

lltl Ilu const(q)

s constt (q, llgllu)

.

Ilgllul-t/q

Ilg-t21111/q

E

Since t I t 2 is in A", Theorem 2.6 is proved.

11

Theorem 2.8. If f is in B1 [resp. D1 , resp, A1] and 11 f IIq < co, q > then f is In 2r [resp. fir, resp. 4r ] for any r in 1 s r < q.

1,

Remark 1. An additional condition is needed to secure that this result is true for r = q (see section 8). Remark 2. The assertion of Theorem 2.8 is not true for r = q, as shown by the following examples.

Example 1. The function f(n) = na if n is a square, and f(n) = 0 otherwise, has norm II f llq = 0 as long as q < 2, and it is (trivially) in ,$1 All RAMANLUAN coefficients ar(f) = M(fcr)/p(r) vanish, but nsx lf(n)l2 ti 'x, and so II f 1122 = M( Ifl2) = 2. But PARSEVAI:s equation M( lfl2) = Y_ cp(r)

lar(f)12 (see section 3) is violated, and f is not in $2.

[This example is due to J.-L. MAUCLAIRE].

Similarly, the function g(n) = -/ log n if n is a prime, else g(n) = 0, has 11g 1l, = 0, all ar(g) = 0, llgll22 = 1, and PARSEVAL's equation is violated again. Example 2. (A. HILDEBRAND). Fix q > 1, and put f(n) = 2klq if n = 2k

is a power of 2, and f(n) = 0 otherwise. Then 'if IIr = 0 if I s r < q, > 0, but f is not in 21q. 11 f Ii q [The proof runs as follows: it is easy to calculate x fq(n) nsx and to show that limx -> m x-1 Ensx fq(n) does not exist (for example, 1

VI 2. properties of q-Almost-Periodic Functions

203

2k+1-1);

let x -4 oo through the sequences 2k and

therefore the mean-

value M(f q) does not exist, and so f9 is not in Al ].

proof of Theorem 2.8. Without loss of generality, let f be real-valued. Define the truncation fK of f by f(n), if If(n)I s K, K, if f(n) > K, -K, if f(n) < - K.

fK(n) =

f E $1 implies that fK E .$1, and - being bounded - the truncation fK is in $4 for every E Z 1. Define s' > 1 by q and fix s by the equas-t+s'-1 = 1. Then, using HoLDER's inequality, tion o(x) = x-1 Ensx If(n) - fK(n)Ir s x 1

SI

'

7-nSx.lf(n)I>K If(n)Ir

iq/(s' q)

nsx

,

JJ

X-1.

l

I

li/s

nsx,lf(n)I>K

Next, Kq

Znsx,lf(n)I>K I S 1nsx,lf(n)1>K If(n)I' 5 (2

II

f IIq)q ' x,

if x is large. Hence, we arrive at lim sup A(x) s II f II q/s ' (2 II f / K ) < E, q q m X if K is chosen large enough, and so f, being near fK E .$r, is in Sr. q/s

II

We state that for real-valued functions f in ,$r the truncated function

fK tends to f in 11. "r' and that, for any E > 0, (2.10)

x-

lim sup x -* o

21

nsx,lf(n)l>K If(n)Ir <

E

if K is sufficiently large. Theorem 2.9 [ DABOUSSI ]. Assume g 2 0, a 2 g°C

E s40 if and only If g E

1,

(i

z

1.

Then

.04°Oa.

The same result is true with A replaced by 2 or B. Corollary 2.10. If g is non-negative and in 4q, where q s 1, then 9jq E '42. The same result is true with A replaced by Z' or S.

Spaces of Arithmetical Function,

204

This corollary comes from Theorem 2.9 with p = 2, a = Zq if q 2 2. If s q < 2, then put f =g' 1; then f2/q = g e 4q, and so 1

f

42.

e

Proof of Theorem 2.9, following DABOUSSI 119801. (1) Assume that g°C c 4p, E > 0. Choose a trigonometric polynomial t in A such that II ga-t 110 s (E/2)a. According to the WEIERSTRASS Approximation Theorem (Theorem A.1.1) there is a polynomial Q with the property I

Q(u) - { max(0,u) }1/a

I

s 2E in IuI 5

IItlI

u.

Then the composition Q ° t is in A [ resp. . or £ in the other cases], and

IIg - Q ° tllaa:E

(2.11)

gives the assertion g c 8°C'3. In order to prove (2.11) we use the inequalities (a)

I x-y Ia ! ,Ix a - ya I in x 2 0, y 2 0,

(b) (c)

(x+y )°C1 s

2«p-i , (x a(3 + ya(i)

in x z 0, y z 0, 1 y-x1, if x and y are real.

l

I max(0,y) - max(0,x) I

S

(b) follows from the convexity of t H tap, (a) is proved utilizing function t H (y+t)a - t of ( without loss of generality x = y+t >

the

y ).

Therefore,

I g(n) - Q(t(n)) lap s { l g(n) - {max(0,t(n))}1/a +

max(0,t(n))

Q(t(n))

p

and using (b), (a) and (c), this becomes s 2ap-1 ,{l g°`(n) - t(n)lp + I { max(0,t(n)) }1/a - Q(t(n)) lap }.

Therefore II

g - Qo t

11,X0 S E.

This is one part of the proof. By Exercise 11 (or Corollary 2.5 (3)) h e

.94k

kc W, implies hk

order to prove the other part, put Y=(

1)

( aP )-1,

a

.a41.

In

V1.2.

properties of q-Almost-Periodic Functions

Then Y

i

1, and

c

205

N. The function h = gl/Y satisfies hY a

[according to the first part of our proof] h

therefore h«PY number c4ly is an integer, and so

a

c

.A4 ap,

.A 'x'3y. The

,A4'. Therefore, go"'

a

.r41 and

the first part of the proof again gives go' E AO . For the question of the existence of a limit distribution of real-valued

functions the following result is useful, as it has already been shown for uniformly-almost-even functions. Theorem 2.11. Let q i 1.

(1) If f E $q is real-valued with values in some finite for infinite] closed interval I = [a, b], and if the function Y': I -- C Is LIPscH[Tz-continuous (so that I W (x) -'F(y)I s L- I x - yI for some constant L > 0), then the composed function 'F ° f is in 3 q again. The result remains true, if S q is replaced by £ I or .4 q. (2) If f e 1q Is complex-valued with values In some finite [or infinite] closed rectangle R, and If the function 'F: R --) C is LipscHiTz-continuous, then the composed function 'F ° f Is in 2q again. The result remains true if $q is replaced by Dq or a4 q. (3) If f In 211 [or 01 or .A41 ] is real-valued then the function nH Is In B1 [or 01 or Al ] for any real t. (4) If q z 1, f E $q, f is real-valued, and infnEN I f(n)1 = S > 0, then f

E Bq.

Proof.

(1) Let s > 0. Choose a trigonometric polynomial t in ,$ [resp. £] near f-tNIIq < e. The values of f are in I; t* is real-valued, without loss f, of generality. If the values of t* are not in the interval I = ]a, b[, replace * t by t = min { b, max(t* a) } (with an obvious interpretation, if a or b are ± a ). t is - nearer to f than t*, therefore II f - t II < E. Then II

II

II

q

q

'1' ° t is even and so in 2 [resp. periodic and so is in £], the values of f and t are in I, 'F Is LIPscHITZ-continuous, and therefore II

'F

f - 'F o t

q

= limX sup x 1 Ex IY'(f(n))-'Y(t(n))Iq n!c

s lim sup x 1. Lq x -* co

nsx

If(n)-t(n)I q s Lq s q.

In the case where f E A q and t ,A4, the function 'F ° t is in s4" by the WEIERSTRASS Theorem, and the proof works in this case, too. E

206

Almost-Periodic and Almost Even Arithmetical Functions

(2) The complex case can be reduced to the real one. Assume that R = Cal, bI] x i Cat, b2]. Then approximate Re f by an even function t

with values in Cal, b1], and Im f by an even function t2 with values in Cat, b2]. The even function t = tI + I' t2 has values in R, and IIf- t IIq s II Ref - tI IIq + II Im f - t2 IIq. The rest may be concluded as in (1).

(3) and (4) are special cases: the functions x H exp (it x ), defined on JR, -)y-1, defined in y Z 8, are LIPSCHITZ-con_ where t is any real number, and y tinuous. Thus 1/IfI E 8q, and f-I = f' IfI-2 E . by Theorem 2.6 (ill).

Examples.

(a) If f is a bounded function in A and P a polynomial with complex coefficients, then the composed function P ° f Is also in A1. This follows from Theorem 2.11, but It could also be deduced from the fact that a bounded function in AI is in A9 for every q 2 1. (b) If f E s4q satisfies b:= sup Re(f(W)) < oo, then exp(f) E Aq. The C; reason is that exp is LIPSCHITZ-continuous In the half-plane {z E

Rezsb}. (c) If f E Aq and a:= inf(Re f(N)) > 0, then log f

a44, because the principal branch of the logarithm function Is LIPSCHITZ-continuous In E

the half-plane {z e C; Re z 2 a} with L = a Remark. If P is an integer-valued polynomial with positive values, for example P(n) = n2+1, then it is a difficult task to prove that f - P Is in Al (or has a mean-value at least) if f is in some A9. The result is not known even for the function µ2, If the degree of P is greater than two.

VI.3. PARSEVAL'S EQUATION

According to section 2 of this chapter the spaces $z C D2 C ,42 are complete vector-spaces with an "inner product"

< f, g> = M(fg). This "Inner product" Is

linear in the first argument; it satisfies

VI.3 parseval's Equation f g>

207

< g, f > and < f, f > z 0, but < f, f > = 0 is possible for

functions f $ 0. Thus, the quotient-spaces modulo null-functions,

2 c D2 c A2, are HILBERT spaces. Theorem 3.1 (PARSEVAL'S equation).

(1) I f f is in

,

2, then Er°1 cp(r)

lar(f)I2 = 11f112

where the ar(f) denote the RAMANUJAN-FOURIER coefficients

ar(f)

M(f cr), r = 1,

= r)

2, ...

.

(ii) If f Is in z2, then Y_

(iii)

r=1 11sasr,gcd(a,r)=1

If f is in

I

M(f ea/r) 12 =

11 f 112

A2, then

«E;R/

12 = 11f112

I

Corollary 3.2.

(i) The set { (p(r))" Cr r = 1, 2, ... is a complete orthonormal system in 22. If f, g are In $2, then }

,

ro 1

(P(r)

(ii) The set { e./r' r =

1,

ar(f) 2, ...,

ar(g) =

1 s a s r, gcd(a,r) =

1

} is a com-

plete orthonormal system in .V2. If f , g are in D2, then Zro

1

lsasr,gcd(a,r)=1 M(f ea/r)

M(g ea/r) = M(f g ).

(iii) If f, g are In A2, then 21 «EW./a

M(g'1«) =

First Proof. The assertions of Corollary 3.2 come from the "Elementary Theory of HILBERT space", which is sketched in Appendix A.2. According

to this theory, the validity of the PARSEVAL equation is equivalent to the denseness [with respect to II. 112 ] of the sets .$, £, A of linear combinations of RAMANUJAN sums [resp. exponential functions] in $2, D2, and A2; this is true by definition of these spaces.

Almost-Periodic and Almost Even Arithmetical Functions

208

VI.4. A SECOND PROOF FOR PARSEVAL'S FORMULA

In this section we present a second proof for PARSEVAL's equation in the space $2. Some properties, perhaps of some Interest, of arithmetical functions in $2 are exhibited, and these properties are used in the proof. Let r be a positive integer, and, for k dividing r, denote by Xk the charac-

teristic function of the set Ak = { n e W: gcd(n,r) = k). Xk

is

a function in r c . (with positive mean-value), therefore , 2 for every f in $2. Consider the linear map F

r : 22

8 r'

fy k 1r

M(xk'

Xk'

This function has the properties given in the following lemma. Lemma 4.1. (1)

F (f) = f if and only if f E 2 .

(2)

If f, g E $2, then M(Fr(f) g ) = M( f Fr(g) ).

(3)

If f e 22, then Fr(f) _ k ak ck, where ak = cP(k) M(f ck).

(4)

If f E $2, g E Br' then II f - Fr(f) 112 s II f - g 112 So Fr(f) Is

(5)

a "best" approximation in $r For every f In 22, the sequence

k1r

AR(f) = Ilf-FR!(f)112' R = 1,

2, ...,

Is monotonically decreasing to zero.

Proof. (1) A function f in 2r is constant on Ak, say equal to d k, for every k dividing r. Therefore, Fr(f) = XkIr dk Xk = f. (2) By definition of Fr and the linearity of the mean-value, we obtain

M(Fr(f) g) = kr M(me) M( Xk g). This expression is symmetric in f and g, and so also is equal to

(3) By the orthogonality of the RAMANUJAN sums the coefficient ak , using (2) and (1), equals

VI.S. An Approximation for 1-Even Functions

{p(k)}-1

ak =

. M ( Fr(f) . ck) _

{9(k)}-1

209

. M(Fr(ck) f) _

{9(k)}-1

. M(ck f).

(4) Without loss of generality, we assume that f and g are real-valued. Let x > 0, klr, and define the function Gxk: IR--jlR, y Hx

1

nsx,n c Ak

( f(n)-y)2.

This function has just one stationary point Xk(n))-1

mx = (2:nsx as x --- ) co,

z

(M(Xk))-1

. M(f Xk) + 0(1)

and this point gives the absolute minimum of Gx k. Therefore,

1.

nsx,n c A. X-1.

=

. (Znsx f(n) - Xk(n)) =

(f(n) - F (f)(n) )2 = r Z

nsx,n c Ak

x-1

(f(n) - (M(X k ))-1 M(f ' Xk)) 2

nsx,n c A.

(f(n) - mx)2 + 0(1) s x-1

I

nsx,n c Ak

( f(n) - g(n))2 + o(1).

Summing over k1r, we find for x -) co IIf - Fr(f)112 s IIf- g 112. 2

(5) At first AR+1(f) s AR(f), by (4). Now, given E > 0, there exists an even function g E . "near" f, 11 f - g 112 < E. Choose an integer R, for which g is in .Y3R. Then IIf -FRI(f) 112 < E,

again by (4).

11

Now we are ready to prove PARSEVAL's equation

ZO1 p(r)

Iar(f)12 = M(If12) =IIf IIZ

For every f E 22, and for every integer R, a standard computation gives rlRl

r

k = 11f 112

2

- rIR! Ep(r)' Iari2.

By (3), the left-hand side is If - FR!(f) 112, which converges to zero by 2 (5).

Almost-Periodic and Almost Even Arithmetical Functloa8

210

VI.S. AN APPROXIMATION FOR FUNCTIONS IN .81

In the last section, the result

112-0, as R -)m, - r was proved by [elementary] HILBERT-space methods. In this section, similar result for arithmetical functions In $1 is given. IIf

u

a

Theorem 5.1 (A. HILDEBRAND). For every function f in St Rm

(5.1)

II

f-

r R1

ar(f) cr IIt = 0, .

where ar(f) = M( f c'), r = 1, 2 JAN coefficients of the function f. {cp(r))-t

,

... denote the RAMANU-

The important feature of this result is that the coefficients of the even functions approximating f are not changed when R is increased. Note that the sequence {R!}R=1,2,... may be substituted by every sequence {nR}R-1

with the property limR---). gcd(nR, r) = r for every integer r.

2

Remark. This theorem allows us to show [again] that the Mornus func6 II µ III = 7t2 > 0, does not belong g to .Y3 t It is known from prime number theory that tion µ, with

.

Md(µ) = limx

x- 1

---> m

'nsx,n=O mod d v(n) = 0

for every integer d: Therefore, (5.2) p(r) . ar(µ) = x

l'

x-1

Zx ti(n) . cr(n) =

dr

d µ(r/d) . Md(µ) = 0.

First we collect some formulae, needed for the proof of Theorem 5.1, as follows. Lemma S.2. (1) For every Integer k, r, R satisfying rIR!, if r.}' k,

0, (5.3)

1 R!

nsR!

n=O (k)

cr(n)

cp(r) k

if rlk. '

V1.5. An Approximation for 1-Even Functions

211

of the

set

= k}, where kJ R! Is supposed. ar(f) for f E 81. Then, for every kIR!, FR! = ZriR! Cr'

Put

k

by

Denote

(2) A

=

{

n E N:

characteristic

the gcd (n, R!) Xk

function

M( f Xk) = M( FR! Xk)

(5.4)

proof. (1) The RAMANUJAN sum Cr is R! - even (and so R! - periodic)

if rIR!, therefore the left-hand side of (S.3) is equal to lim

X --> oo

x-1

c (n) =

nsx,kin r

lim x

d µ(r/d)

dr

= k-1

1

x---> m

nsx,d n,kin

gcd(d,k) .Il(r/d)

dr

IT ( gcd(pZ, k) -

= k-1

1

gcd(pt-1

k)),

pEllr

and this gives the right-hand side of (5.3). (2) FR! is R! - even, and so we obtain lim x-1 X -> m nsx,kln FR!(n) = R!

E

nsR! kin

lim

x -> m

X_

1

x

L

f(m)

a (f)

r

l

c (n)

r

Cr(m) ri R! fp(r)

n R!

R!

Cr(n)

kin =

k-1

lim

x-* m

x-1

msx

f(m)

r!k

Cr(m)'

using (1) and the fact that k divides (R!). The inner sum equal to

rk cr(m) is

k, if kim, (5.S)

n(1+cP(m)+...+

CPt(m)

p"Ilk

0, if k4' m.

Therefore, the functions FR! and f have the same mean-value on the sets Mk = { n EN: kin}, if kI R!, and so also on (S.6)

K=

Mk \ GRIP"*k

MQ.

11

C-0modk

Proof of Theorem 5.1. We start by proving that for any real-valued function f in 231 and every real-valued k-even function g the estimate

Almost-Periodic and Almost Even Arithmetical Functions

212

Ilf-FR,III

(5.7)

kIR!

holds.

FRI and g are R! - even and therefore constant on every set Ak If kIR! (for the definition of Ak see Lemma 5.2 (2)). Denote the values, taken by FR! and g on Ak, by y k I resp. 8k1. Fix k, and assume that Yk z 8k. Then

Zx If(n) - FR,(n)I = Ex (f(n)- yk) + Ex ncA

ncA.

f(n) 2

( Yk- f(n))

ncA,,

f(n) <

iY.

Z nsx

(f(n) - Yk) - nsx 2 (f(n) - Yk), ncA.

ncA,

f(n) 2Y

and, using Lemma 5.2 (2), this is s 2

Z (f(n)- Sk) + o(x) s 2- Z I f(n)-g(n) I + o(x). nsx nsx ncA ncA,,

f(n) aY,

In the other case, yk < 8k, the same estimate Is valid. The sets Ak, kIR!, are a partition of N. Therefore, we obtain nsx

If(n) - F R1 (n)I s

1

Z If(n)-g(n)I + 0(1 nsx

and (5.7) is proved.

To conclude the proof of Theorem 5.1, assume without loss of generality that f e Y31 is real-valued. Given s > 0, choose a real-valued even func-

tion g near f, Ilf -gill < s. If g is k-even, then, according to Ilf-FRI III <2sfor all R2k.

VI.6. LIMIT DISTRIBUTIONS OF ARITHMETICAL FUNCTIONS

For a set a contained in N we define the counting function AN _ 2:

N,ncc

(5.7),

VI.6. Limit Distributions of Arithmetical Functions

213

If f is a real-valued arithmetical function we put FN,a(t) = {A(N)}-1 2:nsN,nca,f(n)st IN we write and in case a = FN(t) = FN IN(t)

for short. As mentioned in Chapter IV, section 3, one says that f has a limit distribution with respect to a if there is a distribution function

F such that limN

. FN,a(t) = F(t)

at every point of continuity of F. An additive function f has a limit distribution, according to P. ERDOS and A.WINTNER's Theorem. [1939], on IN if and only if the three series

-I2 y

1

p,lf(p)I51 P 'f(P), Zp,lf(P)ISjP

'If(P)1

,

-1

P,If(P>i>1 p

converge.

H.DASOUSSI [1981] gives examples of large classes of non-negative additive functions and sets a having no limit distribution. He proved the following theorem.

Theorem 6. 1. Let a c N satisfy (I) limN

>

,,{

A(N)}-1

EnsN,nect,nsO mod d

exists for every d

(ii) to is multiplicative and

c

IN,

P kz

I

where

p-k 1o(pk) < w

If f is a non-negative additive function such P-I,f(p), w(p) +

= d-1 w(d)

Ef(P)>1

that

p-1, w(p) = + -,

XOsf(P)5i

then f does not have a limit distribution on a. More precisely lim

1)

N

co FN. q (t) = 0 for every t.

So F 2 O, F(-co) = 0, F(+c,) = 1, F Is non-decreasing, and continuous from the right.

Almost-Periodic and Almost Even Arithmetical Functioas

214

Corollary 6.2. If a is the set { p+1 } of translates of the primes, then to(d) = d/p(d). Thus, If f Is additive and non-negative and satisfies (6.1) with to = id/cp then f does not have a limit distribution on ,

{ p+1; p prime}.

Proof of Theorem 6.1. The inequalities

(1- a -1) t s 1-e - t s t in 0 s t s 1, (1-e1) s 1-e - t s 1 in t Z I (see Figure VI.3) imply Z

P

P-1'w(p)'(

1-a-f(p)

)=

co.

O.5

Figure

VI.3

Define an additive function fY by truncation: fY(pk) = f(pk) if pk s Y. and fY(pk) = 0 otherwise. The convolution gy = µ *exp(-fY) is multiplica-

tive, IgY(pk)I s 2, and gY(n) = 0 except on a finite set SY of integers, as is seen from the values of g at prime-powers, as well as the relation Y n 1 w(n)'gY(n) = II P-kgY(pk).w(pk)). 1GY = n

wcy

k2

l

The convergence of the series ce of Z

ZpIk22p-k9Y(pk)'w(pk) and the divergen-

oo imply

lim Y - co

P

GY = 0. Next, exp(-fY ) = 1*g, Y

and so

N lim -m

{A(N)}-1

Z exp( -f (n)) = G Y Y nsN,nea

Since exp(-f(n)) s exp-fY(n)), we obtain lim sup {A(N)}-1

N -p _

Z exp( -f(n)) s G -4 0, as y -- co. nsN,nca Y

V1.7. Arithmetical Applications

215

But

{A(N)I-' 'nsN nea exp( -f(n)) = fo a-xdFN Q(x) z e-t. FN Q(t), and Theorem 6.1 is proved.

[positive] results on the existence of limit distributions for additive functions will be given in the next section (see Theorem 7.2).

VL7. ARITHMETICAL APPLICATIONS

VI.7.A. Mean-Values, Limit Distributions.

In number theory the question of the existence of a mean-value is an Important one. Some general mean-value theorems were proved in Chap-

ter II, and a wealth of theorems are known on the existence of meanvalues for special functions, in particular for multiplicative functions. Functions In Ai do have a mean-value. Sections 1 and 2 of this chapter gave many results, producing new functions in Al from "simpler" ones. So these results (Theorems 1.2, 2.3 [(3) - (6)], 2.4, 2.11, Corollaries 2.5, 2.7, ) have consequences on the existence of mean-values, regardless of arithmetical properties such as multiplicativity or additivity. Of course, to make these results useful, it is necessary to provide criteria for functions f to be in A Some criteria of this kind will be given in Chapter VII - but then multiplicativity is relevant. We formulate some examples of results on the existence of mean-values for functions in the spaces considered in this chapter. Proofs need not

be given; they consist of the remark that the function in question is in A', which follows from results given earlier in this chapter.') Al is the largest of the spaces, defined in this chapter. So it is advisable

to formulate conditions ensuring that some resulting function is in ,04.'

Almost-Periodic and Almost-Even Arithmetical Functions

216

(1) If f is In A1, then the mean-values M(f), M(IfI), M(Re(f)), M(Irn(f)) exist.

(2) If f is in Al and g is in ALL [for example, g is a DIRICHLET character, a RAMANUJAN sum, an exponential function a.,, the characteris-

tic function of a residue-class a mod k, where gcd(a,k) =

1, the

characteristic function of the integers prime to [some fixed] k, etc., ...], then the mean-value M(g-f) exists.

(3) If f is in Ai, then the mean-values of the [additively, resp. multipll-

catively] shifted functions fn H f(n+a), f(b-) : n H As long as n+a s 0, put f(n+a) =

1

where

for accuracy.]

) exists (a e Z); if q, q' > 1, q-1 + q'_1 f in A q, g E A q, then M( f gia+i) exists.

(4) If f, g E A 2, then M( f

g(a+)

(S) If fl, ..., fk are in Ak, and If bX > 0, ax are integers

=1,

(x = 1,...,k),

then the function F: n H f1(bI n+ al)

...

f k ( bk n + ak)

has a mean-value.

[This is a generalization of L. LucHT's results; this author only dealt with multiplicative functions, but he obtained product formulae for the mean-values (see L. LuCHT [1979a, 1979b]). The continuity theorem for DIRICHLET series (see the Appendix) might be helpful in calculating the mean-value of the function F given in (5) in the case of multiplica-

tive functions, but some additional conditions seem to be necessary to obtain "nice" results.] (6) If f is real-valued and is in ,94q, where q 2 1, and if the image f(N) is contained in a closed Interval I c IR, and if `1: I - C is LIPSCHITZ continuous, then the composed function To f is in .4q, and so it has a mean-value.

Examples for 'Y are the functions z '- z-1, z '- exp(z), z y log(z), etc. Of course, one has to be careful about f(N), and some assumptions on the values of f are necessary before (6) or other versions of Theorem 2.11 are applicable.

VI.7. Arithmetical Applications

(a) Let q 2 1. If f then 1/ f .v4 q.

E

217

34 q is real-valued, and if infnfiN If(n)I = 8 > 0,

E

(b) If f e .1q is complex-valued, and if supnEiRe(f(n)) s K < co, then exp(f) E Aq. (c) If IF

E A q is complex-valued, and if Inf... Re (f(n)) > 8 > 0, then

log IF E Aq.

The calculation of the mean-value can (given appropriate circumstances) be dealt with by an application of the continuity theorem for DIRICHLET series:

Theorem 7.1. If f: IN -i C has a mean-value M(f), then

M(f) = lima-),

(7.1)

1+

n=t f(n)'n

In particular, if f is multiplicative, then the calculation of the limit (7.1) often is rather simple. Proof. The existence of the limit M(f) Implies

I f(n) =

o(x), as x --3 oo.

nsx

Partial summation gives, as long as d >

Ex f(n)-n-' = Ex f(n)

.

x-' - f1

1,

Yu f(n)

du,

and so

f

o(u),u-°-1

du

1

= M(f) o

(0-1)-1

+

0((o-1)-t)

as o -4 1+.

The asymptotic relation i;(o) = ((j-l) + 0((0-1) ), as o -4 1+, gives the assertion.

According to the continuity theorem for characteristic functions (see Chapter IV, section 3) the question of the existence of a limit distribution in the sense of probability theory is a problem of the existence (and continuity) of certain mean-values. We prove the following theorem.

Theorem 7.2. If g Is a real-valued arithmetical function in .s41, then there

Is a limit distribution for g; this means that the limit N-1 . n{n s N; g(n) s x lim (x)

N-m

a

218

Almost-Periodic and Almost Even Arithmetical Function,

exists In the sense of probability theory.

For the proof it has to be shown that the mean-value Mt = M(n H exp{ itg(n)})

(7.2)

exists for any real t, and that the function t H Mt is continuous at t = 0. According to Theorem 2.11, the function n H exp{ itg(n)} Is in 011; hence the mean-values Mt exist. The continuity of t H Mt follows from the estimate x-1 (eltg(n) _ I) s lim sup x 1 Z lim nsxll x -- m / x -3 M nsx ,

=

Itl

IIg111.

VI.7.B. Applications to Power-Series with Multiplicative Coefficients.

Given an arithmetical function f, the region of analyticity for the generating power-series (7.3)

F(z) = 1'=i f(n)

.

zn

may be of some interest. In some sense "most" power series with radius of convergence equal to 1 are non-continuable across the unit disc in the complex plane [see, for example, L. BIEBERBACH [19SS]]. Of course,

a number theorist would like to obtain an answer to the question of non-continuability of F(z) if the coefficients of this power series are arithmetical functions with some arithmetical property. G. POLYA and G. SZEGO's Theorems [see BIEBERBACH [1955]] state:

if the coefficients f(n) of the power series (7.3) with radius of convergence equal to one are integers [ resp. assume at most finitely many

distinct values], then either F represents a rational function or it

is

analytically non-continuable beyond the unit circle.

For multiplicative arithmetical functions L. LuCHT and F. TUTTAS [1979] proved the following result.

V1.7. Arithmetical Applications

219

If f is a multiplicative function with finite (semi-]norm II f Ill, and if the mean- value M(f) exists and is non-zero, then the power series F(z) = I '=i f(n) z° is non-continuable beyond Its circle of convergence if and only if f(p k-1)

9(p )

$

vak

for Infinitely many prime-powers pk. Otherwise F(z) represents a rational function.

This theorem relates special properties of the coefficients of the power series to the global behaviour of the function represented by this series. and Z'=1 For example, the power series Y -'=i are non-continuable. The LUCHT-TUTTAS condition is, in fact, a condition related with the RAMANLUAN coefficients of the arithmetical function f. We are going to show that the property "Multlpllcativlty" does not play an essential role; more important is that the RAMANLUAN coefficients ar(f) _ {p(r))-1 M(f c) do not vanish "too often". Theorem 7.3. Let f e 82.

(i) If Infinitely many of the RAMANUTAN-(FoURIERJ-coefficients ar(f)

{(p(r)}-1 M(f

Cr) are non-zero, then F(z) _ I 'f(n)-z' is

non-continuable beyond the unit circle.

(ii) If only finitely many coefficients ar(f) are non-zero, and if f Is represented (pointwisel by its RAMANUTAN expansion, f(n)

ar(f)

cr(n), for n = 1,

2, ...,

then the power-series F(z) = -n=t f(n) zn represents a rational function.

Remark 1. By HILDBBRAND's Theorem (V, Theorem 1.2) the RAMANLUAN

expansion is convergent to the correct values f(n) if f is in 2u. Later we shall show that the same is true for multiplicative functions in 212, supposed that M(f) $ 0 (see VIII, Theorem 5.1). Remark 2. Using formulae for the RAMANLUAN coefficients, which will

be deduced in Chapter VIII (see VIII, Theorem 4.4), it is easy to show that in case of multiplicative functions the non-vanishing condition of

Almost-Periodic and Almost Even Arithmetical Functiop$

220

infinitely many RAMANLUAN coefficients is equivalent to LUCHT's condition given above.

Remark 3. The assumption f e $2 may be replaced by f e 21. The proof has to be changed in so far as PARSEVAi s equation has to be replaced by a result by A. HILDEBRAND, proved in section S (Theorem S.O.

Remark 4. Differentiation does not destroy the property of being rational

or non-continuable beyond the unit circle. Therefore, the result can easily be extended by replacing the assumption f e $2 with:

There is some non-negative integer k such that n H Example. Theorem 7.3 is no longer true if f f

e Ou. This may be seen from the function

f,

n-k. f(n) is in 22.

$2 is replaced by

e

defined by the uni-

formly convergent series f(n) = G1sk<m

This function is in Lu; the power series n=1

f(n) zn =

Zk

2

exp (2ni/k z )

1 - exp( 2iti/k ) z } -1

k=1

is continuable beyond the unit circle, but it is not a rational function. Proof of Theorem 7.3. PARSEVAL's equation gives (7.4)

II

f - lisrsR ar(f)' Cr

1122 = Er>R (p(r)' Jar(f)12 < E

if R z R.(E) is sufficiently large. The generating power series for the function G1srsR

is R(Z) _

n=1 1 2:1srsR ar(f)'cr(n) }

= 2: 1srsR ar(f)

Zm

. z"

mod r WZ .

(1-mz)-1,

where m runs through the primitive rth roots of unity, m =mgr =

gcd(a,r) = 1.

This function R(z) is a rational function and is "near" the following sense: IZ

n=1

f(n)'zn - 9{(z)

I

< 2E

' (

1-IzI

)-1

n=1

f(n)-z' in

VI.7. Arithmetical Applications

221

if Jzi < 1 is near 1. This may be seen from (7.4), using partial summation.

Therefore, if z = t. ma r, 0 < t < 1, t -) 1-, and if ar(f) $ 0, then I

2:-n=1

f(n)-z'

and so mar is a singular point for F(z). But if ar(f) 4 0 infinitely often,

then the corresponding points mar are dense on the unit circle, and the non-continuability of F(z) is proved. [The asserted denseness of the points mar may be deduced from a Theorem from CH. HOOLEY [Acta Arithm. 8, 1963], given as follows. Theorem 7.4 (CH. HOOLEY). Denote by I = al < a2 <...
s r the

]

VL7.C. Power Series Bounded on the Negative Real Axis

An interesting problem was posed by L. RUBEL and K. STOLARSKY [1980]. These authors sought the determination of all subsets N1 C IN with the property that the power series neRV xn/n.

shares with the exponential function the property of being bounded on the negative real axis. The solution of this question by these two mathematicians shows that there are, besides 0 and IN, exactly four subsets INl with the property mentioned above.

Analogously, we seek the determination of all multiplicative or additive functions f in 2 (with the additional assumption M(f) $ 0 in the multiplicative case) for which the function

Ef() =

f(n)

n=l

i

W!_

zn

is bounded on the negative real axis. We prove the following theorem.

Theorem 7.5. (a) Let f e $2 be a multiplicative function with meanvalue M(f) $ 0, which is represented by Its RAMANIIJAN expansion,

and assume that Ef(z) Is bounded on the negative real axis. Then

Almost-Periodic and Almost Even Arithmetical Functions

222

f = 1 or f Is periodic with period 4.

More exactly, with some complex parameter c, the function f

is

given by 1,

f(n)

1-c,

=

1+c,

if n = I or 3 mod 4, If n = 2 mod 4, If n 0 mod 4.

The corresponding functions E. W are

e' - I - c ( cos x + 1 ), c c C. (b) If g E S2 Is additive, if g is represented by its RAMANUJAN expansion, and if E (z) is bounded on the negative real axis, then g g(n) _

c,

If 41 n,

-c,

if 211 n,

if

0,

The corresponding functions E

are

E (z) = c ( cos z 9

Remark. The assumption, that f (or g) is represented by its RAMANWAN expansion, is automatically true for multiplicative or additive functions in 32 (see Chapter VIII).

For the proof of Theorem 7.5 consider the Laplace transform (this is called the "Borel transform" in the theory of entire functions) 9 f(z) = fO Ef(-t) e-tz dt = Z -=I (-1)n f(n) z-n i The integral representation together with the boundedness of Ef(-t) in t 2 0 shows that the Laplace transform.? f(z) is holomorphic in Re z > 0, and the second representation as a power series implies (using partial summation and'nsx f(n) = O(x) or the estimate f(n) << -/n, which comes from 11f112< co) that 9f(z) is holomorphic in jzi > 1. Therefore, the power series

-

z

:n-1 f(n) zn

V1.7. Arithmetical Applications

223

continuable beyond the unit circle; according to our results in the preceding section, this implies that at most finitely many of the RAMAis

NUJAN coefficients of f are non-zero. The function f is represented by its RAMANU7AN expansion, therefore the rationality of the power series

follows and f is in 2, and so f(n) = E1srsR ar Cr(n).

Inserting this formula into the power series expansion of - f (-Z-') we obtain, after a short calculation, the exponential polynomial Ef(z) = Iisr, R ar 2]lsasr,gcd(a,r)=1 l exp ((J.

1 1.

Boundedness on the negative real axis Implies ar = 0 for every exponent having real part Re(wa,r ) < 0 [for the details of = exp( w this this argument see the paper by RUBEL & STOLARSKY 119801].

For r * 1, 4, 6 there are primitive roots of unity w with Re(w) < 02) therefore ar = 0 unless r = I or 4 or 6. We shall show in Chapter VIII, is multiplicative. ThereTheorem 4.4, that the function r H fore, a6(f) = 0 because a2(f) = 0. Next, 1 = f(1) = a1c1(1) + a4c4(1) = a1,

and all the possible solutions of our problem are f(n) = 1 + a4c4(n);

these are indeed solutions, and the multiplicative case is settled.

In the additive case we know (this will be shown later) that ar = 0, if

r is not a power of a prime, therefore in this case a6 = 0, too. The value f(1) is zero, and so a1 = f(l) = 0, and all the solutions are given by f(n) =

2)

If r23 Is odd, then gcd(2(r+1),r) 1, and w = do. If 4lr and r a 8, then take w = exp(27t1 (1 r+1)/r). If r+2)/r). then take w =

will 211r,

r 2 10,

Almost-Periodic and Almost-Even Arithmetical Functions

224

VI. B. A Sq-CRITERION

The condition f c Sq, where q 2 1, implies f E S1 and II f 11q

< co-

Is the

reverse assertion true? It follows from Theorem 2.8 that f E S 1 and < oo for some q lead to f E Sr for 1 s r < q, but f E Sq is not II f II true in general. In his dissertation (1988, Frankfurt; parts of this dissertation are published as [1989a, b, c]) P. KUNTH proved the following theorem.

Theorem 8.1. For every q > 1 the conditions (8.1) and (8.2) are equivalent. (8.1)

f E Sq

(8.2)

(a) f E S (b) II f Ii q < (c) lim r - q-

II f II

= II f IIq.

The same theorem holds (with the same proof) for the spaces and £q instead of $q.

.r4q

Remarks.

(1) In his proof, P. KUNTH used tools from functional analysis centering around the concept of uniform convexity. The proof given here uses standard approximating techniques. (2) For every arithmetical function f, the function r

-

IILII, [1, co[ -9[0, cot,

is non-decreasing. (8.2) (c) means that this function is semi-continu-

ous (from the left) at the point r = q. (3) Condition (8.2) (c) is clearly equivalent with (8.2) (c*)

limr---> q _ II f IIr = II f IIq .

If (c) or (c*) Is violated, then

f E Sq is not true.

II f IIs = oo

for any s > q because

225

VI. B. A 2q -Criterion

proposition 8.2. For q > and every arithmetical function f f E Bq if and only if f e 21 and IfI E 2q. 1

The same assertion is true for the spaces £ 1, A q.

proof. The implication from left to right is contained in Theorem 2.3. So, let f e .$1, Ifl a $q be given. We factorize f: f=g

( f ), where g = max 11,

l f I }.

g

The first factor g is in $q for 1 E ,$q IfI a $q. Since g z 1, by Theo$1. Therefore, the second factor (f/g) is in ,

rem 2.11 we obtain (1/g) .

a

1; it is bounded, and Theorem 2.6 gives f e $q.

proof of the easy implication (8.1) will be performed in two steps.

(8.2). It suffices to prove (c). This

Every bounded function f c $1 has property

(8.3)

q - r > 0, K = sup,. IN lf(n)l. Then, if

Proof of (8.3). Let q > r z 1, x > 0,

A = E I lf(n)l - lf(n)lr1 = E lf(n)lr I If(n)IS -1 I nsx nsx

+

E nsx

If(n)IS -1 I.

1
O
Using

S'Ilogyl,

if 0
S' l logy l ys, if y Z

1,

we obtain As

E nsx

If(n)Ir Ilog If(n)II S+ E

nsx

If(n)Iq .log If(n)I

S

1
O
Hence,

Ilfllq sc 8 +I1 fl1r,andIIfIIr5c S+Ilfllq. Therefore,

I

II f 11q - II f Ilr

i

s c S = c ( q - r), and (8.3), and so (8.2)

for bounded functions is proved.

Almost-Periodic and Almost Even Arithmetical Functions

226

Figure VI.4 gives the functions ys - 1 (lower

0,09 0,08

curve) and S logy ys (upper curve), where

0,07 0,06

0.05

in the range

0,04

8 = 0.1,

0,09

1 s y s 2.3.

0,02 0,01

F 1 g u r e VI.4

In the second step we show (8.2) (c), if f e $q. Without loss of generality, we assume that f k 0 ( f e $q implies IfI Sq), and use the truncated function E

fK = min { f, K)

.

From section 2 (see (2.10)) we know that fK E $q for any K > 0, and that limK---> Of _fK Ilq = 0. If 1 s r s q, the Inequality 0 s 11f llq - 11f Ilr S (Ilfllq - IlfKllq) + ( IlfKllq - IlfKllr) + (IlfKllr - 11f I1r)

s2

Ilf-fKllq + (IlfKllq - IlfKllr)

holds. Given s > 0, we find K > 0 such that 2

11 f - f K llq

<

''

2

E,

and

(by (8.3)) a real number ro E [1, q[ with the property I1fK11g- IlfKllr < 2 E for every r in [ro,q]. For these r we obtain II f Ilq - II f 1Ir < E, and (c) is proved.

Proof of the implication (8.2)

(8.1) in three steps. Let q > 1.

For every e > 0 there exists a real number r0 E [1, q[ with the property: for every sequence (an)n=1,2.... with 0 s an s 1 the Inequality

(8.4)

0 S Z (arn - aqn) < S. X nsx

holds for all x > 0, r

e

[ro,q[.

order to show (8.4), put S = q - r, where q > r. The function h(x) = xr - xq takes a maximal value in the intervall [0,1] at the point xo = )1/5, and the maximal value is h(xo ) = ( r )r18 ( S ) s S. ( 1 ) In

,

.

q

q

q

q

227

V1.8. A 2q -Criterion

Therefore, there exists a So > 0 such that the maximal value of h is less than s for every 8 E 10,80 [. The desired inequality is correct for r = ro := q - So, and by monotonicity for every r in [ro,q[. For any f $1, f 2 0, with II f Ilq < co and s > 0 there exists a real number r o E [1,q[ such that c

(8.5)

0s Ilf-fKllq - Ilf - fKllrs r IIfIIq- IlfiVr +E for all K >0 and all r in [ r o, q [ .

proof. The difference f(n)- fK(n) = 0 if f(n) s K, and = f(n)-K if f(n)

>

Using (8.4)

K.

and the monotonicity of x H xq -

xr in

[1, oo [, we calculate for K > 0, x z 1, Y_

nsx

(If(n) - fK(n) Iq - If(n) - fK(nlr) = 21 + E20

where 0

(f(n)-K)r - (f(n)-K)q)

X n f,,

X

2E

Ksf(n)sK+1

for all r E [ro, q [, and EZ =

I

nsx, f(n)>K+1

I

nsx,f(n)21

((f(n)-K)q - (f(n)-K)r)s

(fq(n) - fr(n)) =

Z

nsx, f(n)>K+1

(fq(n) _fr(n))

E (fq(n) - fr(n)) + R,

nsx

where

0sR=

I

nsx,O
(

fr(n) - fq(n)) <

by (8.4), for every r c [ro,q[. So, for all x z

0 s x-1 . z2 s X . Znsx fq(n) 1

x-1

1,

.

nsx fr(n) +

1 e.

Since f ,$r (where 1 s r < q) and f > 0, Theorem 2.9 gives and the mean-value of fr exists; we obtain E

lim sup.

fr

E

1

x 1 2 s IIfIIq - IIfIIr + 2E

Collating our estimates, we obtain the estimate x-1

znSx If(n)-fK(n)I1 s

x-1

znsx If(n)-fK(n)Ir +

x-1.12

Almost-Periodic and Almost Even Arithmetical Functions

228

for all K > 0, r E [ro,q[, x Z 1. For x

oo this implies

Ilf - fKIIgSIIf -fKII + UfIIq-IIfiir+£, and (8.5) is proved.

Now the missing implication in Theorem 8.1

is

easily proved. Given

E .i with II f IIq < oo, and llmr q- IIf Ilr = II f IIq we assume that f > 0 without loss of generality because of Proposition 8.2. Having f

,

chosen s > 0, we find a real number ro E [1, q[ by (8.5) such that

Ilf-fKIlgsllf-fKllr+IIfllq-IIfIIr+3E for all K > 0 and r [ro,q[. Choose r such that II f IIq - IIfIIr < 3 E, and then K > 0 so that I f - f K Ilr < 3 E. Then II f - fK 11q < e. The E

function fK is in Sq, and so Theorem 8.1 is proved.

Remark. We give a second proof for the more difficult implication (8.2) of Theorem 8.1, using DABOUSSI's Theorem 2.9. Given with (8.1) (c*) and f 11q < co, we assume f z 0 without loss of generality because of Proposition 8.2. For every r, 1 s r < q, and x > 0, (8.1)

fE

2I

we see that Z nsx

(f'q(n) - f2r(n))2 = Z f' (n) + fr(n) - 2 Z fi (q+r)(n). nsx nsx nsx

By Theorem 2.8, f is in ,fir; DABOUSSI's Theorem gives fr E 2 ), and so

the mean-value M(fr) = M( f (q+r)

f Ilr

exists. The same argument applies to

Therefore, II f,

- f'rll22= IIfIiqq + IIfoorr - 2

IIf1I5cq+r)

(q+r)'

Making use of (C) we obtain lim

r-> q-

Ilf3q _ frllZ = 0.

f is in Sr, therefore f'r E S2 (again by Theorem 2.9). Being approximated

by functions in 22, the function f'q itself is in ,$2. Using Theorem 2.9 once more, the function f is in Sq.

229

VI.9. Exercises

VI.9. EXERCISES

1) Give a [simple] direct proof for the fact that arithmetical functions in A 1 have a mean-value.

2) If f : IN -4 IR is an integer-valued function in $", then f is in $. Give an integer-valued function in 21 which is not in B.

3) Denote by ADD resp. ADDS the set of additive [resp. strongly additive] functions. Prove that these are subspaces of CIN, and that the

II1 - completion of (ADD n,1) [resp. of (ADDS n,l) ] is a subspace of ,1. II

4) Assume that f e D1 is a non-negative multiplicative arithmetical function. Denote by MP (f) the limit lima -> x-1 nsx Pin f(n). Prove that for every prime Zk21 P_k f(pk) < co if and only if

MP (f) $ M(f). 5) Let f be a multiplicative function in 01. For every prime power p k prove

lima

(a)

-

m

x-1. 1nsx,P`IIn f(n) = p-k. f(pk) (M(f) - MP (f) ), x-1

pf(p,) ( M(f)- MP(f)), Gnsx,Pn f(n) = Iekk if the series on the right-hand side converges absolutely.

(b)

lima

6) Prove Theorem 2.11 (3) directly.

7) Let y > 0 be an irrational number. Denote by g(n) the number of positive Integers m with the property Cy m ] = n. Prove:

(a) g is in A2 (b) Put S = y-1 - [y-11. Then the FOURIER coefficients of the func-

tion g are g (a) = y-1, if a = 0,

g(a)21ciaY)

1

(e

and g(a) = 0 otherwise.

27cI ocy (8-1)_

-1

7L,a$0,

Almost-Even and Almost-Periodic Arithmetical Functiops

230

(c) What does PARSEVAL's equation mean? Answer:

X'=1

I

n-2. sln2(rt8n) =

,7E2

(S - S2 ), where 0 s 8 < 1.

8) Give a proof for PARSEVAL's equation in ,Z2, using methods similar

to those used in section 4. Hint: Ak = {n E IN; n = k mod r},

9) If f is in

S1

Fr(f) =

and S > 0, then the function h, h(n) =

f (n) If(n)If 8-1

Jf(n)j > 8,

f(n), if If(n)Is 8,

belongs to ,$1. .V1

and every residue-class s mod r the mean-value limx -i M X-1 nsx, n a s mod r f(n) exists. Prove this result for coprime r, s, using the formula

10) For every function f E

lnsx, n - s mod r f(n) = {P(r)}-1 ZX mod r (X(s)' Ensx X(n) f(n) ). 11) If q1 > 1, ..., qk > 1, q1-1 + ... + qk-1 =1, and f1 E .44+,

then prove that the product f1

f2

...

fk E

4 k,

fk Is in A1.

12) Let klr, where k and r are positive integers. Calculate the meanvalue of the indicator-function of the set {neN; gcd(n, r) = k}.

Photographs of Mathematicians

231

A

E. WIRSING

r H. DABOUSSI

R. RANICIN

I

P. D. T. A. ELLLIOTT

H. DELANGE

A. RENYI (1921-1970)

232

Photographs of Mathematicians {

Jk

$

A. SELBERG

I

1

M. JUTILA & M. N. HUXLEY

H. E. RICHERT

A. KARACUBA

C. L. SIEGEL

A. Ivic

(1896-1981)

J.-L. MAUCLAIRE

M. NAIR

233

Chapter VII

The Theorems of ELLIOTT and DABoussi

ABSTRACT. This chapter deals with multiplicative arithmetical functions

f, and relations between the values of these functions taken at prime powers, and the almost periodic behaviour of f. More exactly, we prove that the convergence of four series, summing the values of f at primes, respectively prime powers [with appropriate weights], implies that f is in 2q, and (If in addition the mean-value M(f) is supposed to be non-zero) vice versa. For this part of the proof we use an approach due to H. DELANGE and H. DABOUSSi 119762 in the special case where q = 2; the general case is reduced to this special case using the properties of spaces of almost-periodic functions obtained In Chapter VI. Finally, DABOUSS7's characterization of multiplicative functions in ,A4q with non-empty spectrum is deduced.

The Theorems of Elliott and Daboussi

234

V I I.1. INTRODUCTION

As shown in the preceding chapter, q-almost-even and q-almost-periodic

functions have nice and interesting properties; for example, there are mean-value results for these functions (see VI.7) results concerning

the existence of limit distributions and some results on the global behaviour of power series with almost-even coefficients. These results seem to provide sufficient motivation in the search for a, hopefully,

rather simple characterization of functions belonging to the spaces s44 D Z) q D $4 of almost-periodic functions, defined in VI.1. Of course, in number theory we look for functions having some distinguishing arithmetical properties, and the most common of these properties are additivity and multiplicativity. According to the heuristics outlined in Chapter 111.1, conditions character-

izing membership of an arithmetical function to, say, X34, ought to be formulated using the values of f at primes and prime powers.

Historically, theorems of this kind were given for the first time in connection with the problem of the characterization of multiplicative functions with a non-zero mean-value. The E. WIRSING Theorem, proved

in 11.4, is an example of the fact that assumptions about the behaviour

in the mean of values of a multiplicative function, taken at primes, imply asymptotic formulae for the sum Z f(n). But these results do ns x not characterize multiplicative functions with a non-zero mean-value. In 1961, H. DELANGE proved the following theorem.

Theorem 1.1. Let f: N -) C be a multiplicative function satisfying IfI s 1. Then the following conditions are equivalent: (1.1)

The mean-value M(f) = lim x-I Z f(n) exists and is non-zero. X --) nsx (i)

The series S1(f)

(1.2)

=

21

p-I (f(p) - 1) is convergent,

P

(ii)

Osk< m

p

0 for all primes p.

Introduction

23S

Remark. The assumption IfI s 1 Implies that

2 P kf(pk)I for every prime p 2 3. Therefore, as did DELANGE, the validity of (1.211) is to be assumed only for p = 2, and it may be substituted by IEosk<.

the DELANGE condition f(2 k)

$ -1 for some k k 1.

In 196S A. RENYI gave a simple proof of the implication ( (1.2)

(1.1)),

using the TURAN-KUBILIUS inequality (see 1.4). This method of proof

will be the basis of the more general result, given as Proposition 3.2 in this chapter. The condition IfI 5 1 was removed by P. D. T. A. ELLIOTT in 1975, who replaced this severe restriction by the assumption II f IIq< 00, with the semi-norm II f 11q defined In VI, (1.3). We define the [ELLIOTT-] set Cq of multiplicative functions- f: EN -4 C by the following conditions:

Definition 1.2. f E f'q If and only if

(I) the DELANGE series SI(f) = E p-I ( f(p) - 1) Is [conditionally] P

convergent, (li) the series

S2'(f) =

S

P, If(p)I

p-I

S/4

I

f(p) - 1

12,

and

S2,q (f) =

p-I

p, If(p)I > 5/4

If(P)Iq

are convergent,

(iii) the series

S3,q (f) =pEkk2 E P-k If(pk)Iq is convergent.

Remarks. 1) The series SI(f) is conditionally convergent, the primes being ordered canonically according to their size. The other series are 'absolutely convergent.

2) In the special case where q = 2, condition (ii) is equivalent to the convergence of the series (ii')

S2(f) = E P-1 P

.

I f(p) - 1 I2.

The Theorems of Elliott and Daboussi

236

Using this notation, P. D. T. A. ELLIOTT [197S] proved the following theorem.

Theorem 1.3. Assume that f : N -* C is a multiplicative function, and assume that q > 1. Then the following conditions are equivalent. and the mean-value M(f) exists and is non-zero.

< oo

(1.3)

II f II

(1.4)

f Is in 9 q and condition (1.211) is satisfied,

In

9

this chapter we are going to show that the convergence of the

series in Definition 1.2 implies, in fact, that the multiplicative function f is in 8q (Theorem 4.1). Furthermore, following DABOUSSI and DELAN-

GE, we prove (Theorem 5.1) that for any multiplicative function f with mean-value M(f) $ 0 the following properties are equivalent:

Finally, we characterize multiplicative functions in Aq, possessing a non-void spectrum (see Theorem 6.1). We begin with some rather simple consequences of the condition If

II f II

9

II f 11q< oo.

< oo, then there exists some positive constant c such that If(n)I s c nt"q for every n E IN,

(1.5)

and [by partial summation from nSx If(n)I s C x ]

1 n_t n-' If(n)Iq < oo, if Re s > Lemma 1.4. I f

I I f1 1

q

< oo for some q > 1, then, E p

X Z

1.

I

f(p) z p

< co, and

p-k. lf(pk)Ir < co for every r In I s r < q.

p k22 In particular, using the notation of Chapter III, Section 1, a multiplicative arithmetical function f, satisfying II f 11q < oo, belongs to the

set

III I.1. Introduction

237

{f:IN - C, f multiplicative, Z I

f(P) I2< oo, 2:

p-k

1:

If(pk)I < m}.

p kz2

p

P

proof. Choose an e > 0 such that I + 2 E < q. HOLDER'S inequality and (1.5)

imply P

x

I ff(p) 12 5

p

I (n)k

c.

I

p2 - (2+e)/q

Psx S C .(

-LAY)j p(1+e)

Psx

\1/q (

q' \1/q 1 { Psx l p2 (2+e)/q j J

By (1.6) and the choice of s, both series on the right converge for x - -co. Similarly, with s > 0, 1+2E < q the estimate ,

p-k(1+e),If(Pk)Iglr/q.(2i z p-k(1-E a_ )`1 q

E

p kz2

p kz2 psx

/

P cx

p k22 p"sx

proves the convergence of the second series. Example. The following example shows that an extension of Lemma 1.4 to r = q is not possible. Define a multiplicative function f by f(pk) = 0

if p > 2 or k Is odd, and f(2k) = (t-1 , 22e)1/q if k = 2 Then I I f IIq = 0, but X 21

p kz2

l;

is even.

p-k. If(Pk) I q = 21 2-k , fq(2k) kz2

Lemma I.S. Let q > 1, f: W - C be multiplicative,

II f II

g

= Do.

< oo, and assume

that the mean-value M(f) exists and is non-zero. Then there exists a prime p 0 with the properties M(P)(f) =x limes

(1)

x-1 ns n

f(n) = M(f)'

{pf(p,1)}-

for every prime p z p0, and (2)

M

(d)

(f) =

x lim -. -

x-1

nsx,(n,d)=1

f(n) = M(f) j7{ W (p l) Pld

}-1

f

for every positive integer d which consists only of primes p Z po.

(3) M (f) = P

lim x-1 x -+ -

nsx, n=O mod p

f(n) = M(f) {pf(p,1)-1} {(pf(P,1)}-1

Remark. If f is 2-multiplicative, so that f(pk) = 0 for every k Z 2, then

The Theorems of Elliott and Daboussl

238

the mean-values in question are given by

M(d)(f) = M(f) II

M(P) (f) = M(f) { 1 + P-1' f(p) }-1,

P

fppl

MP ( f ) = M

Proof. In Re s 2

1,

{I+

P-1. f(p)

Id

{ 1+p-1 f(p)

(1.5) implies

Ik

p-ks , f(pk) 15 c

{ pt - 1/q - 1} 1,

1

therefore there is some p0 such that [recalling the abbreviation pf(P,s) = 1 + P-'-f(p) + for every prime p 2 po, pf(p,s)

in Re s z

C. 1.

{ pi - 1/q - 1

}-1

z 1 - C.

{ PD

1/q -

1

1

1}

2z

Let p* be a fixed prime greater than or equal to p0,

Define a multiplicative function g by

f(Pk), if p * p",

if p = p

0,

.

functions f and g are related, f is in ;, IgI s Ifl, therefore g e §. For every prime p Z p0 the factor