Gap and Density Theorems (Colloquium Publications (Amer Mathematical Soc))

.1 t:.

r /7' \ \ ;i1 '

V

:RO TYPE

[IX]

. A~ )

, ell.cn ) "

E'log A2

'

.. m

•. _[

- C27 q,'(N) e·(N).

1

Am2

- eU(n)

)

, e2.(N+l») 1 -eU(n) -­

\

177

PROOF OF THE MAIN THEOREM

[31 ]

11 - (~(N~~ k)21 ~ (~ + ~) logj ~

(1

n1

26q,'(N)e·(N)(q,(N)

+ 1)

2

+ [n log udu _ 2aq,'(N)q,(N)e'HN) _ 26q,'(N)e",(N)

~ nl log nl + n2log ~ - nl - n2 - 2aq,'(N)q,(N)e·(N) - 2aq,'(N)e...(N) ~ nl log n1 + n2 log n2 - 26q,'(N)q,(N)e"' CN ) - 46q,'(N)i(N). For fixed N, nl + n2 is fixed. By differentiating x log x + (c - x) log (c - x), 1 ~ x ~ c - 1, )

it follows easily that it takes its least value for x = 1.

1 - e-2(n-N-1).'(n»)

nl log nl

+ n2 log n2 ~

Thus using (31.13),

(2&P'(N)l(N) - 2) log (2&p'(N)e"'(N) - 2).

Therefore for large N

1 - e-n.'(n»). ,",' log £.oJ

11 -

A~+ k)2 ~­ 26q,'(N)e...(N) log (26q,'(N)e",(N) 1

(ewn

- 2)

- 26q,'(N)q,(N)e.(N) - 6q,'(N)e·(N)

: (1 - e·... (n»)

~

2aq,'(N)e...(N) {log q,'(N)e... CN ) + log (26 - 2e-",(N)/q,'(N») I - 26q,'(N)q,(N)e"'(N) - 6q,'(N)e·(N)

~

26q,'(N)e·(N) log q,'(N) - lOq,'(N)e·(N).

Using this and (31.12), (31.11) becomes log I F'(A m )

(31.14)

I ~ 26

(E + f) q,'(n)e·(n) 1

+

.'(N)e",(N) , :~

I

< q,(u). indicated to cover

1

N)e·(N) log (2e·(N»). ~er

values from - n1

11 - e~) I n

26q,'(N)e"' CN ) log q,'(N) - C30 q,'(N)e",(N).

By the same proof as used in obtaining Lemma 30.2, in somewhat simpler form since here a(u) = 1, it follows that

(E + f:) q,'(n)e·(n) og (2e,HN»)

log

N+l

(31.15)

N+l

log)

1- eUA~ I= !q,'(N Cn )

+ !q,'(N + l)e",(N+l) log /1

+ (f

1

N1 - +

-

l)e", cN - 1) log

A~_

eU(N 1)

I

eU~;+1) I

1"" ) q,'(u)e"'Cu ) log 11 - eA~ N+l

11 _

.

2",(u)

I du

+ O(q,'(N)e· CN»).

If the right side of (31.15) is handled by the same method used in the proof of . Lemma 29.1 with the very great simplification that here a(u) is not involved, (31.15) becomes

>,

",

...

"1 ,

,\

"

.~~

'l~ .'

\

[XI

[331

A series

theorems on the gamma and zeta functions.

(T), for large u

..,

Ea 1

,nd it is this that !.O7) is essential. tly much weaker ~(x), (32.07) will all for the usual B the much more

J.l"e-""''''

"

1 - e-""''''

is known as a Lambert series. Thus a corollary of Theorem LUI is that the higher indices theorem holds for Lambert series. This resu'lt was unknown before the proof of Theorem LU in 1937. In fact the Hardy-Littlewood higher indices theorem remained an essentially isolated theorem until the theorem for K(x) was proved. We shall also prove the following result. THEOREM LIV. 5 Let the hypothesis of Theorem LI be satisfied. let xK(x) f L( - 00, 00). Then (32.14)

It can be shown -plane v ~ 0, so lere for the case

lim inf {f(x) - sJ

"'... ..,

=-

lim sup {f(x) - s}

11,

= 11

implies that

(32.15)

lim sup

I

t

s

ak -

1

I~

Clll

where Cl is an absolute constant depending only on L and K(x). if An+l - An ~ 00 tJwn Cl = 1.

>

O.

om well-known

In particular,

33. Reduction to a lemma on biorthogonal functions. The proof of Theorem LI can be at once reduced to the proof of the following lemma. C2 , Cs , ... denote positive constants depending only on Land K(x). LEMMA 33.1. If the hypothesis of Theorem LI is satisfied, there exists a sequence of functions IRn(x, B) I, (n ~ 1), such that

1:

), 00) and N'(x)

for which

In addition

","'00

n-+oo

L

191

REDUCTION TO LEMMA ON BIORTHOGONAL FUNCTIONS

(33.01) (33.02)

Rn(x, B)

1 1:

Xn/2

(33.03)

I Rn(x, B) I dx < C2, I Rn(x, B) I dx

-00

(33.04)

Rn(x, B) dx

and

(33.05)

;i~

1:

=

0,

x

C ~ ~,

A"

1:", K(y) dy = 0,

Rn(x, B) dx

>

CsB,

n

~

1,

k ¢ n,

1:-", K(y) dy = 1.

Before proving the lemma we shall use it to obtain Theorem LI. • For K(x) = e-<"'e'" this theorem was proved by Ingham, On the higher indices theorem oj Hardy and Littlewood, Quarterly Journal of Mathematics, vol. 8 (1937), p. 1.

Iii

ill iii iii iii iii

,

?""

-{

...~

~,-:JP"1 ,

\:'.

•

/

". ,{}

il i !I I'! III III Pi

:Ii

,
~II!

...

I', :1

II'

ii

ii II' ii ',I,il\

[XI

lim sup I n-UO

L: I

yK(y) Idy.

t

ak - s I

1

~

lim sup If(lI,,) - s I fl.-GO

+ lim I t

1

n-'CO

34.1.

For every n

>

0 there exists an H,,(s)

H,,(A,,) = 1;

(34.01)

H,,(Ak) = 0,

where {A,,} are as in Theorem LI. (34.02) then h,,(u)

h,,(u)

= 0, I u I > c,

E

ak - f(lI,,) I =

n.

L( -

00,

00),

sw:,h that

k rf n,

=

h,,(u)

L:

'Iii

ill'I'

Ii!

iii'II

i!i

ii II: ii,

g"(u)e-iu~,,,

111 ill

ii!

I gn(U) I <

0, let

(Jk

= =

(Jk

=

(Jk

Ak).

C6

I g~(u) I <

,

I:i

C6

•

This defines

IUk}

+ An I > }-L for all m > 0, tkL if tkL - Am + An = tL for some m, Am - An if -}-L ~ tkL - Am + An < tL.

lkL if I lkL - Am

uniquely and

I.

I Uk

-

tkL

I~

}-L.

~

}-L.

Also Uk+1 -

I ud

Uk

includes IAm - A,,} for all m rf

T(s) =

n.

IT (1 - ~)(1 - ~). Uk

1

U-k

Then T(s) I sin 21rs/L

Clearly

ill

!I!

i!IIi

H,,(s)e-iOUds,

Proof of Lemma 34.1. Throughout this proof n is some fixed positive integer. For - 00 < k < 00 let

Thus

il

il

(34.04)

Moreover Let

Iiilli

'I'

where

r>

II,

:Ii

I" ii,

and

(34.03)

iii

Iii

If

= (2:)1 /2

il!

II;

I!I

This proves C 1 = 1 when A,,+1 - A" ~ 00. 34. Proof of the lemma. We now turn to the proof of Lemma 33.1. We will require several auxiliary results. The first of these is an interpolation result. LEMMA

ihat s = 0 since, mes

195

PROOF OF THE LEMMA

[ 34]

I=

-~

21r1 s j

IT 111 - - 2s/kL 1111 +- 2s/kL I. S/Uk

1

S/U_k

111!

Ill! il ; jill"

Ii

.~.

"

~1(

"

.,

i i!

!

ii

"on ,,",~

iil i:

il!'

\/ I 1•••••••••••••••••••••••••••••••.il~••••••••••••• \

l-'

\,

,~/

~

:!II

11 ,Iii

illl

lil

,II!

iili

r ilil iiil :111

,ii

[XI t

. ~ + ~i(-v-+l) dJ;. 'nic function in the d~

t 341

I r,,(u, B) I ~ Thus for 1 u

For 1 u \

I

to X(u, v).

Let

CoBI

~

U

Il(u)

u+C

1

u-c

I cjJ(By) Idy

jB(U+C)

1

~ Col U Il(u) B(u--c)

1

I cjJ(y) Idy,

B)

~

I

L:

Col u le6 (2C)

> 2c and B > 2 it follows from 1

r,,(u, B)

6

Thus

L:

(34.19)

I

I cjJ(y) I dy

~

C16 1 u I·

.

_

I lill 'III !,!I!I

cjJ(y) dy ~ 1

(ul

r,,(u, B) 1 du

<

'I'

C17

+u

2'

il

i~

i'l

~I

I!I

ClB •

Ilil

i'l'1l i!I'

il!

illl

iuB

= -,-

ifII

1

From (34.16) we also have .!:..-le'ul.,.r,,(u, B)J du

'1

III

(34.08) that

f

~ C5 1 u le (u)

I

:,i'1 ,

'ii

2c it follows from (34.08) that

I r,,(u,

(34.18) ~te

IIII :1 1

(34.17)

+r

199

PROOF OF THE LEMMA

"'n'"

fU+C v-c

k(y)cjJ(By)g~(u - y)iA""dy

iB ( u/c1(U)) (211") 1/2 k(u) , 1 - k(u)

1 u

+c

v-c

I i!i!

k(y)cjJ(By)g..(u - y)e'>''''' dy.

l!il

il,1

Treating each of the terms on the right in a manner similar to that used on (34.16), it follows easily that of e-g(u).

Thus for

len

I:u {eM"r..(u, B)} I ~ 1 ~u 2 '

(34.20)

u

:'lli

'ill

i!

j!~

iii

If we define R..(x, B)

= (2:)1/2

L:

r,,(u, B)e'u," dU,

i'l r; l!i

it follows from (34.19) that (34.21)

I

R,,(x, B)

1

<

C18

• ,.'

follows. Let <1>(8) d. (34.10) follows

Also

;

,.

R,.(x

+ A", B)

= (2:)1/ 2

L:

= (2:)1/ 2

L. d~

r,.(u, B)c'V>"e'U'" du,

ments of Theorem

..

and integrating by parts,

,

iA""dy.

~

!!~

I:! u

-ixR,,(x

+ An, B)

{r,,(u, B)e'''>''JeiV:< du.

U

I 1;1

By the theorem for Fourier transforms, Theorem E, and (34.20)

:f,j [i!

:i!l lj~

ii

n

;:'~