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plainly satisfied with k > 0.
7. t Bohr, 7 cf. Ill, J Bohr, 2, 5. In a memoir as yet unpublished.
Hardy and Littlewood,
:

(1)
These conditions are rather wider than that adopted by Schnee and Landau, It is natural to suppose substantially the same as that adopted by Bohr.
and are h and k it is
a convex function of K. shown by Bohr If that the conditions
also
H
Bohr, 6.
2.
MULTIPLICATION
61
necessary and sufficient that a given sequence
Summation
13.
of Dirichlet's
by other methods.
Series
the
It
is
natural to enquire whether methods of summation different in principle from those which we have considered may not be useful in the theory. The first to suggest itself is Borel's exponential method. The application of this method to ordinary Dirichlet's series has been considered by Hardy and by It has
Feketef.
been shown, for example, that the regions of summability, and that the method at once
and of absolute summability, are halfplanes
;
gives the analytical continuation all over the plane of certain interesting But the method is not one which seems likely to render classes of series.
great services to the general theory. RieszJ has considered methods of
summation related to Borel's, and its as the typical means of this somewhat generalisation by MittagLeffler, These methods lead to section are related to Ces&ro's original means. the function associated with the series which differ one very important respect from those afforded by the fundamentally Their domains of application may, like Borel's means. of typical theory polygon of summability, or MittagLeffler's etoile, be defined simply by
representations of in
of the singular points of the function, and necessarily contain singular points on their frontier.
means
VIII
THE MULTIPLICATION OF DIRICHLET'S SERIES
We
shall be occupied in this section with the study of a special problem, interesting on account of the variety and elegance of the 1.
results to
which
in the Analytic *
it
has
led,
Theory
of
The construction given by Bohr
abscissae
(I.e.
.
pp. 127 et seq.), for a series with given
may be simplified by using the series "2e
element* in place of the series
t Hardy, 3; Fekete, J Riesz, 6. In this connection seq.
and important on account of its applications
Numbers
AA
'
^ ri~ 8
of IV,
4, (5)
as a
'
simple
which he uses.
1.
we
refer particularly to
Landau, 4, and H., pp. 750
et
62
MULTIPLICATION
We denote by A C the
and by
B the series
and
'
'
productseries
+
G!
C2
I
...,
a function of the a's and
to be defined more precisely A, B, C to denote the sums of the series, when they are convergent or summable. When C is formed in accordance with Cauchy's rule*, we have
where
c w is
moment.
in a
We
cp
=a
shall also use
1
^ + a ^_ 2
+
1
+a^j^
...
infinity.
power is
are led to m
series 2,a m
x
it
^b n xn
,
2
am bn
.
m+np+l.
however, only one among an by arranging the formal product of the
Cauchy's rule for multiplication
We
6's,
in
is,
powers of x and putting
x 1,
or,
what
the same thing, by arranging the formal product of the Dirichlet's
series
^ // m e/?>"
,
ns s h t>
m
+ n, associating according to the ascending order of the sums has the same value, and then together all the terms for which
mn
putting s
=
conception
0.
It is clear that
of multiplication
we
arrive at a generalisation of our
by considering the general Dirichlet's
series
Sa^X
*b n e**'
and arranging their formal product according to the ascending order Let (vp ) be the ascending sequence formed by of the sums A w + /x n Then the series (7 = 2^, where all the values of A m + /^f. = ^ tt m O n Cp .
,
^m^Mn =
will
/
f
P
be called the Dirichlefs product of the series A, jB, of type (X, /x). if X w = log m, p n = log w, so that we are dealing with ordinary
Thus
Dirichlet's series, then v p
=
cp
=
log/>
and
2 a m bn =2a d bp d /
the latter summation extending to *
See
Bromwich,
all
the divisors
d
of p.
Infinite series, p. 83.
generally the case in applications that the \ and /* sequences are the Any case can be formally reduced to this case by regarding all the numbers and pn as forming one sequence and attributing to each series a number of
t It same.
\m
e.g.
,
d
inn=p
is
terms with zero coefficients (Landau, //., p. 750). In the most important cases = logm) the v sequence is also the same, but of course this is not (e.g. X m =w, Xm generally true. In the theoretically general case no two values of X m f /* n will be equal.
MULTIPLICATION 2.
63
The
three classical theorems relating to ordinary multiplication (Cauchy's, Mertens', and Abel's) have their analogues in the general theory.
THEOREM absolutely
53.
A
If
B
and
This theorem is merely a special case of the which asserts that the absolutely convergent double be summed indifferently in any manner we please*.
THEOREM
54.
A
//
C
are absolutely convergent, then
is
AB = C.
convergent and
is
absolutely convergent
classical
series
theorem
2 a m b n may
B convergent,
and
then
C is convergent and AB = C. t We shall prove that 2 a m b n converges to the sum A B when
arranged as a simple series so that a m b n comes before a m>b n if X w + n < \ m + n ^ p (the order of the terms for which \ m + /* has the same value being >
,
,
Theorem 54 then follows by bracketing all the terms for has the same value. Suppose first that B = 0. Let Sv be any partial sum of the new series, and let a k be the a of highest rank that occurs in it. Then
indifferent).
which \ m +
where r
ay by
/x n
a function of k and p. Suppose that Then it contains all the terms
.
is
O=
<*pb q
Thus k S y and
Now we
r
Sy
for jp
=
7,
8V
contains a term
1, 2, ... y.
can choose y so that
2 ap <
and
.
\

y+i
Then
fifj
<
2 ap + 

M2 a
where
A
denotes the
sum
of the series
greater than the greatest value of is
to say as A * GO *
See
e.g.
p
y+i
1

Br
. \
2 ap and Thus Sv *
M
is
any number
as y * GO
,
that
.
Bromwich,
Infinite series, p. 81.
This theorem
is
not merely a special
Theorem 54, because it asserts the absolute convergence of the product series. t Stieltjes, 2 Landau, 4, and H., p. 752. See also Wigert, 1. J This is the kernel of the proof. The reader will find that a figure will help to
case of
;
elucidate the argument.
64
MULTIPLICATION
b\
We
*0.
Secondly, suppose
=
bi
~
B>
'
#2
bs
,
= 68
,
. .
..
converges to zero, and so
2<
AS.
converges to
THEOREM
3.
=
b%
Then, by what precedes, 2,a m b n
form a new series B' for which
55.
the
If
series
A, B,
C
are all convergent,
AB^C. We the analogue of Abel's theorem for power series*. it from a more general theorem, the analogue for Dirichlet's series of a wellknown theorem of Ceskrot. This
is
deduce
shall
56. If A is summable (A, a) and summable (v, a + /? + 1), and A B = (7.
THEOREM then
C is
=a+
If
B
is
summabte (X
/3),
we have
/?4l
the summations being limited respectively by the inequalities X m <
<w
"
)
<w

For consider the term
and
o>,
Then
a^.
It
occurs in
C yv
(t*>)
Aw
if
t/x H
i),
its coefficient is
(o>A m /An)y.
The term a m and bn occurs
Hence
occurs in in
B^ (o>
a^fe^ occurs
A^ T) if
(T) p. n
if
Xm <
<w
r,
with coefficient (r  Xm ) a T with coefficient
T,
/*JP.
(CD
on the righthand side of
(1) if
Am
4 p. n
<
,
o>,
and
its
coefficient is
*
The theorem was
given by Landau, 4, in the case in which at least one "* ** nS 6n e of the series S a m e **, possesses a region of absolute convergence. His of function theory. A purely arithmetic and proof depended on considerations completely general proof was then discovered independently by Phragme'n, Riesz, first
" AW
This proof depends on the particular case of Theorem 56 in which See Landau, H., pp. 762, 904 ; Riesz, 2 Bohr, 2. t Cesaro, 1 ; see also Bromwich, Infinite series, p. 316. Cesaro and Bromwich The extension to nonintegral consider only integral orders of summability. orders is due to Knopp, a, and Chapman, 1.
and Bohr.
a=/3=0.
;
MULTIPLICATION
Thus
and
But
(1) is established.
therefore,
Lemma
by
65
5,
This proves the theorem. In particular, vergent, the product series is snmmable (v, follows from
The
4.
Theorems 56 and
THEOREM then
(cf.
is
57.
If
summable
A
(v,
is ft)
1).
16.
following generalisation of
companion theorem to Theorem
and B are conTheorem 55 then
A
if
Theorem 54 provides an
absolutely convergent,
and
B
is
summable
is
evidently enough
.#
= 0.
(/x,
/3),
and C=AB.*
In this theorem the Xsequence is at our disposal. It 2) to prove the theorem in the particular case when
We
interesting
56.
have 2
C(o>)= There
w
a constant
is
for all values of T
;
a TO Mfi>A m /!)*=
2 A m
M such that
and we can choose
side of this inequality can be replaced
M
so that (1) the on the righthand by if rw >i
Then we have
cD'^CC^O:
and so which proves the theorem.
The next theorem which we
5.
shall state is one
analogous to those of the 'Tauberian' theorems of VI, Theorem 35. We therefore omit the proof.
whose general idea is 7, and in particular
*
For the special case of multiplication in accordance with Cauchy's rule, Hardy and Littlevvood, 2 (Theorem 35), where further theorems on the The particular theorem proved there is multiplication of series will be found.
see
however a special case of one given previously by Fekete, 8. H.
&
R.
5
MULTIPLICATION
66
THEOREM
58.
If
am _/)
Awi^wA
(
A and B
then the convergence of
Our
6.
last
(2)
is
r + r'X),
2am
senes
ensure that of (7.*
to
enough
different character.
If
T>0, r'0, tfAe
(^
7
bn _/)/f*n~J*n
)>
two theorems are of a
THEOREM 59 1. (1)
xm
* e~ Am
i*
p + T>p', convergent
+ r'>p, = p, cwd absolutely for p'
convergent
s=p + T,
for v
(3)
tfAe
series
for s=p' + r'
2&n e~ MnS w
s
convergent for
= p',
ewc
absolutely convergent
;
^Ac?i the series
2cp e"~ VpS
is
convergent for >
pr
__ "~
'
i
h
i
p r f rr
'
r+T
We
shall give a proof of this theorem only in the simplest interesting case, viz. that in which
X w =logm,
/i n
and most
v p =logp,
=logw,
so that the series are ordinary Dirichlet's series, and
p=p'=0.
We
can then suppose that r and T are any numbers greater than
may
A
be any number greater than
and
B are
The theorem
.
1,
so that
therefore asserts that,
if
convergent, and
=
Cp
then 2c p p~ 9
is
# 7n Oft
2,
mnp
convergent for cr>^.
,
In this case, however,
it is
possible
to prove rather more. * This
theorem
ensure that
cp
=
is
not a corollary of Theorem 35.
The conditions do not The theorem was proved, in the particular case and in the general case by Hardy, 7. Hardy
{("p^pi)/^p}
\m = m, n n n> by Hardy, 2 however supposed the indices X w ;
Xm
~
,
n subject to the conditions
ytt
\nl =
(X
m
),
A*

Mi =
(M n )
.
That these conditions are unnecessary was shown by Rosenblatt, 2. t Landau, 4, and #., p. 755.
MULTIPLICATION
THEOREM
If A and
60.
B
67
are convergent, then 2 
is
convergent*.
We
shall prove, in fact, that 2c p~* is p uniformly convergent along finite stretch of the line ti.
any
s=f
A x *= 2
Let us write
m>x
and similarly
for
J5.
We
am
,
have
00
2
Similarly
and these
includes
bv
/
v~*=o
{
1 \
7
relations all hold uniformly as regards
all
t.
We observe now that
8 products of pairs of terms a m m~ b n n~* for which T?W ,
Vr V# 2 am m~ 8 x 2 i
6 n ?i~ 8
m<
cc
a
< [#], and
i
which Ja, n < V^ ; and that, not greater than ^/#. It follows that
all for TI is
:
)
W?V
n
m/i
if
< [#],
one at least of
m arid
V v^  2 am m~* 2 ^?i~ 8 1
i
= V2
*/w
VaJ
/wi
am m~* 2
6,,i/"*4
2 b n n~* 2 a v
i/~
8
1 v'or
,
l
)
xm
2 i
1
,
V^
=o(l);t
which proves the theorem. It was suggested by CahenJ that the convergence of A and B should involve the convergence of Sc p p~* for (r>0, and not merely for o^i (as is shown by Theorem 60). This question, the answer to which remained for long doubtful, was ultimately decided by Landau , who showed by an example that Cahen's hypothesis was untrue. *
Stieltjes, 1, 2.
t Since
See also Landau, 4, and xhn S b v v" B = o(x
Jf.,
pp. 759 et seq.
V# x/n
and similarly
^a v v~ = o(x
,
8
*).
Va;
J Cahen, 1.
Landau, 5, and H.,
p. 773.
52
68
MULTIPLICATION
This may be seen very simply by means of Bohr's example (III, 7) of a = 1 a for 00, for which (0) square this function, we obtain a function for which p. (cr) = 2 2
if o<. It follows from Theorem 12 that the squared series cannot converge for
so that ft(cr)>l
Bohr, 5.
BIBLIOGRAPHY [The following list of memoirs does not profess to be exhaustive. It contains (1) memoirs actually referred to in the text, (2) memoirs which have appeared since the publication of Landau's Handbuch (in 1909) and which are concerned with the theory of summable series or the general
We have added a few representative theory of Dirichlet's series. memoirs concerned with applications of the idea of summability in allied theories
such as the theory of Fourier's
series.]
N. H. Abel *
Untersuchungen
(1 )
Math., vol.
1,
iiber die
Reihe
^ \
^1
1
1826, pp. 311339 ((Euvres, vol.
F. R. *
x
1 f
.
^ 2t 1,
vol. 9,
. . .
',
Journal fur
pp. 219250).
Berwald
Solution nouvelle d'un probl&me de Fourier',
(1)
x* 4
Arkiv for Matematik,
1913, no. 14, pp. 118.
H. Bohr Sur
l
(1) (2)
la sdrie
de Dirichlet', Comptes Rendus, 11 Jan. 1909.
'tfber die Summabilitat Dirichletscher Reihen', Gottinger Nachrichten, 1909, pp. 247262.
Sur
4
(3)
la
convergence des sdries de Dirichlet
',
Comptes Rendus,
1
Aug.
1910. '
(4)
Beweis der Existenz Dirichletscher Reiheri, die Nullstellen mit beliebig grosser Abszisse besitzen pp. 235243.
(5)
'Bidrag
til
'*,
Rendiconti
di Palermo,
vol.
31,
1910,
de Dirichlet'ske Rcekkers Theori', Dissertation, Copenhagen,
1910.
Reihen ', die Summabilitatsgrenzgerade der Dirichletschen Wiener Sitzungsberichte, vol. 119, 1910, pp. 13911397. 'Losung des absoluten Konvergenzproblems einer allgemeinen Klasse
'
(6)
(7)
Uber
Dirichletscher Reihen
'
(8)
'
(9)
7
Acta Math., vol. 36, 1911, pp. 197240. , tJber die gleichmassige Konvergenz Dirichletscher Reihen ', Journal 203211. fiir Math., vol. 143, 1913, pp. tTber die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen 2a n n~ 8 ', Gottinger Nachrichten, 1913, pp. 441488.
BIBLIOGRAPHY
70 Bemerkungen
(10) 'Einige
Reihen
*
(11)
Reihe
?,an n~* als
Math, und Phys.,
Konvergenzproblem
Funktion der Koeffizienten der Reihe', Archiv der
ser. 2, vol. 21,
1913, pp. 326330.
H. Bohr and E. Landau (s) und fK (5) in der Na'he der Geraden
(1)
Oottinger Nachrichten, 1910, pp. 303330. 'Em Satz iiber die Dirichletschen Reihen
fFunktion und die ZFunktionen 1913, pp. 269272.
the limits of certain infinite series and integrals 1908, pp. 350369. relation between the convergence of series
Lond. Math.
Soc., ser. 2, vol. 6, 1908, pp.
E. 'Sur de
la fonction ((s) I'ficole
Normale,
',
vol. 37,
',
Math. Ann.,
and of
integrals
vol. 65,
',
Proc.
327338.
Cahen
de Riemann et ser.
1
mit Anwendung auf die
Rendiconti di Palermo,
On
The
'
(2)
',
FA. Bromwich
T. J. '
(1)
(1)
Dirichletscher
tiber das Verbal ten von f
'
(2)
das
liber
Rendiconti di Palermo, vol. 37, 1913, pp. 116. Darstellung der gloichmassigen Konvergenzabszisse einer Dirichletschen ',
3, vol.
des fonctions analogues', Ann. 11, 1894, pp. 75164. stir
E. Ces&ro *
Sur
(1)
la multiplication des series',
Bulletin des Sciences Math., vol. 14,
1890, pp. 114120.
S. *
On
'
On
(1)
(2)
Chapman
nonintegral orders of summability of series and integrals', Proc. Lond. Math. Soc., ser. 2, vol. 9, 1910, pp. 369409.
the general theory of summability, with applications to Fourier's
and other
(3)
series ', Quarterly Journal, vol. 43, 1912, pp. 153. the 'On summability of series of Legendre's functions', Math. Ann., vol. 72, 1912, pp. 211227. '
(4)
Some theorems on
the multiplication of series which are infinite in both directions', Quarterly Journal, vol. 44, 1913, pp. 219233. (See also
G. H.
Hardy G.
(1)
'"fiber
die
Holderschen
and S.
Chapman.)
Paber
und Ces&roschen
Grenzwerte
',
Munchener
Sitzungsberichte, vol. 43, 1913, pp. 519531.
P. (1)
Fatou
'Series trigonom^triques et series de Taylor', Acta Math., vol. 30, 1906,
pp. 335400 (These, Paris, 1907).
BIBLIOGRAPHY
71
L. Fej6r *
Untersuchungen uber Fouriersche Reihen', Math. Ann.,
vol. 58, 1904, pp. 5169. '"Ober die Laplacesche Reihe', Math. Ann., vol. 67, 1909, pp. 76109. ' La convergence sur son cercle de convergence d'une se'rie de puissances
(1)
(2) (3)
effectuant
une representation conforme du
cercle sur le plan simple Comptes Rendus, 6 Jan. 1913. 'Uber die Konvergenz der Potenzreihen auf der Konvergenzgrenze in Fallen der konforrnen Abbildung auf die schlichte Ebene', //. A. Schwarz Festschrift, 1914, pp. 4253.
(4)
',
M. Pekete Sur les series de Dirichlet', Comptes Rendus, 25 April 1910. 'Sur un th(3oreme de M. Landau Comptes Rendus, 22 Aug. 1910. A szettart6 vegtelen sorok elrneletehez ', Mathematikai es Terme'szettudo
*
(1)
J
(2)
,
'
(3)
mdnyi (4)
firtesito, vol. 29,
1911, pp. 719726.
'Sur quelques generalisations d'un thcoreme de Weierstrass Comptes Rendus, 21 Aug. 1911. Sur une propriete des racines des moyerines arifchm^tiques d'une s^rie entiere reelle', Comptes Rendus, 13 Oct. 1913. f Vizsgdlatok az absolut summabilis sorokr61, alkalmazdssal a Dirichlete's ',
'
(5)
(6)
Fouriersorokra
',
Mathematikai
es
Terme'szettudomdnyi Ertesito,
vol. 32,
1914*
(1)
M. Pekete and G. Polya 'Uber ein Problem von Laguerre', Rendiconti di Palermo,
vol. 34, 1912,
pp. 89120.
W. (1)
B.
Ford
between the sumformulas of Holder and Cesaro American Journal of Math., vol. 32, 1910, pp. 315326.
'On the
relation
',
M. Fujiwara (1)
'On the convergenceabscissa Math. Journal,
vol. 6,
Dirichlet's
series',
Ttihoku
1914, pp. 140142.
T. H. (1)
of general
Gronwall
(2)
Math. Ann.,
vol. 75, 1914, pp.
A. (1)
321375.
Haar
't)ber die Legendresche Reihe', Rendiconti di Palermo, vol. 32, 1911,
pp. 132142. *
We
are unable to complete this reference at present.
72
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'Sur
les series
Hadamard
de Dirichlet',
Rendiconti di Palermo,
vol.
25,
1908,
pp. 326330, 395396.
H. Hankel '
Die Euler'schen Integrate bei unbeschrankter Variabilitat des Argumentes', Zeitschrift filr Math., Jahrgang 9, 1864, pp. 121.
(1)
O. H. (1)
'On
certain
oscillating
Hardy Quarterly
series',
vol.
Journal,
38,
1907,
Proc.
Lond.
pp. 269288. (2)
'The multiplication of conditionally convergent Math.
Soc.,
ser.
2, vol. 6, 1908, pp.
series',
410423.
(4)
'The application to Dirichlet's series of Borel's exponential method of summation Proc. Lond. Math. Soc., ser. 2, vol. 8, 1909, pp. 277294. 'Theorems relating to the convergence and summability of slowly
(5)
pp. 301320. On a case of termbyterm integration of an infinite series ', Messenger
(3)
',
oscillating c
of Math.,
series',
Proc.
vol. 39, 1910, pp.
Lond.
Proc. Lond. Math. Soc., ser.
The
'
(7)
ser.
Soc.,
2,
vol.
8,
1909,
136139.
Theorems connected with Maclaurin's
'
(6)
test for the
convergence of series ',
1910, pp. 126144. multiplication of Dirichlet's series ', Proc. Lond. Math. Soc., ser. 2,
vol. 10, 1911, pp. (8)
Math.
vol. 9,
2,
396405.
'An
extension of a theorem on oscillating series', Proc. Lond. Math. Soc., ser. 2, vol. 12, 1912, pp. 174180.
On
'
(9)
the summability of Fourier's series 365372.
',
Proc. Lond. Math. Soc., ser. 2,
vol. 12, 1912, pp.
(10)
'Note on Lambert's
series', Proc.
Lond. Math.
Soc., ser. 2, vol. 13, 1913,
pp. 192198.
G. H. (1)
'A
(1)
'The
Hardy
Q. H.
Hardy
and
relations between Borel's
Proc. Lond. Math. Soc., ser. '
(2)
(3)
and S.
general view of the theory of vol. 42, 1911, pp. 181216.
2,
Chapman
summable
J.
series',
B. Littlewood
and Ces&ro's methods of summation', vol.
11, 1911, pp.
Contributions to the arithmetic theory of series ser. 2, vol. 11, 1912, pp. 411478.
'Sur
la
s^rie S)
Quarterly Journal,
',
116.
Proc. Lond. Math. Soc.,
de Fourier d'une fonction & carre soinmable', Comptes
28 April 1913.
BIBLIOGRAPHY (4)
'Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive', Proc. Lond. Math. Soc., ser. 2, vol.
(5)
13, 1913, pp. 174191.
'Some theorems concerning vol. 43, 1914, pp.
Dirichlet's
series',
Messenger of Math.,
134147.
New
proofs of the primenumber theorem and similar theorems', Quarterly Journal, vol. 46, 1915 (unpublished). Contributions to the theory of the Riemann zotafunction and the
(
(6) '
(7)
theory of the distribution of primes
',
Acta Math, (unpublished).
Harnack
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