332
J. P. TULL
(April
References
1. P. Erdos and S. J. Taylor, On the set of points of convergence of a lacunary
trigonometric series and the equidistribution properties of related sequences, Proc. London
Math. Soc. (2) vol. 7 (1957) pp. 598-615.
2. C. S. Herz, The Bohr spectrum of bounded functions,
Bull. Amer. Math.
Soc.
vol. 62 (1955) p. 76.
3. H. Weyl, Uber die Gleichverteilung von Zahlen mod Eins, Math. Ann. vol. 77
(1915-1916) pp. 313-352.
Cambridge,
England
THE MULTIPLICATION PROBLEM FOR
DIRICHLET SERIES
J. P. TULL
E. Landau [l, §214] has given a theorem on the multiplication
of
Dirichlet series to the effect that if a, P, p, t, are real numbers with
min (p, r)>max
(a, P) and if 22a"£»s converges for cr>a, absolutely
for cr>p, 22c''"£nS converges for a>P, absoutely for o>t, then the
Dirichlet product of these two series converges for
or — aP
cj >
-
p + t —a —P
•
(If min (p, t) ^ max (a, P) then
we have
convergence
for
cr>max (a, P).) H. Bohr [2, Theorem XIX] gave an example to
show that in the case a=/3 = 0, p=r = l the above conclusion cannot
be improved.
In this paper we shall use a variation of Bohr's example to give,
for each a, P, p, r with min (p, r)>max
(a, P), two Dirichlet series
whose product has abscissa of convergence exactly
pr — aP
p + t — a —P
Thus we show that Landau's theorem is the best possible in all cases
(the trivial cases being handled similarly).
Bohr [2, Theorem XVII] defines a certain Dirichlet series 22amw_*
as follows. Let (an), (tn), (Pn), (jn) be sequences of positive integers
such that for all re^ 1
Received by the editors September
16, 1957 and, in revised form, September
1957.
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23,
1958]
THE MULTIPLICATION PROBLEM FOR DIRICHLET SERIES
333
2
a„ < tn < fin < yn < «n+l>
fin = tH ",
where
an <
Qn)112,
lim 5„ = 0,
t„,
lim t„ " = 0.
n
For example,
7n >
n
we could take
t„ = 223", Sn = 2-2",
fin = 223"+2",
Let c be a given positive
number
a„ = 22""1 -
yn = 223"+i + 1.
and define Sm= E^-i
0
(1)
1,
ai by
for an fk m < fin,
Sm = ■m"'»
for fin^m
1
^ yn,
for 7„ < m < an+i.
(In Bohr's original work c = l). Since |5m| ^1 for all m and the sequence (Sm) has no limit, it is clear that the series Ea»»w_s has convergence abscissa 0. Thus the abscissa of absolute convergence is at
most 1, and so p(a) =0 for <r2:1, where p. is the Lindelof function for
f(s) —Ea"»W-
Bohr shows for c = l, 0<cr<l,
throughout
Bohr's proof we replace
0<o-0<l,
as «—>oo,
(2)
/((TO +
tC<„) = -
that p(a)^l
—a. If
tn by ctn we will find that
tn
+
0(tn
for
).
(TO
(Hence,
ju(o-)S: 1 —o- for 0<<r<l;
actually,
from
[l, §229], we can
show, with Bohr, that p(o) = 1 —a for 0<o-<l).
Now, given a<p,
g(5) = /(-)
\p — a/
Then for a<aa<p,
take c = (p—a)~1 and let
= E
a'r^m, ',
|J, = m
since 0<(o-0— a)/(p— a) <l,
/<ro — a
(3)
where
g(ao + tin) = /(-h
Vp_tt
t
t-1
tn
" ,
am = aJim.
by (2), as »—><*>
\
P_a/
l_(,ro_a)/(p-a)(l+8„)
= tn
ao— a
+ 0[t„
Similarly, given fi<r, take c = (r—fi)~1 and let
Then for fi<o-0<T,
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l_(jc_a)/(p_„)(l+jnl
1.
334
J. P. TULL
(4)
A(cr0 + t*B) = -
/»
If max (a, /3)<cr0<min
,
, ., .,,
g(0-O + ttn)h((T0
+ Oj 4
(p, r), then
, . x
+ tin) = "-—-T
}.
by (3) and (4), as re—>oo
_1
2-l(»0-a)/(p-a)+((ro-/3)/(T-S)l(H-JB)
tn
(<70— a) (do — P)
+
Thus
the Lindelof
function
, 2-((<r(r-a)/(p-a)+(<roHS)/(T-B)l(l+«B)1
0{tn
for gh satisfies,
(cr — a
ti*)H2-\-+- \p — a
in this interval,
\ .
since /„5n—>0,
cr — P)
t — PJ
and so ju(cr) > 1 for
cr < (pr - aP)/(p + t - a-
P).
Observe that
pr — ap
max (a, /3) <-
< min (p, t).
p + r — a -p
Therefore, by [l, §229], the Dirichlet product of g and h cannot converge if a<(pr —aP)/(p+r
—a—P), and so the abscissa of convergence is exactly (pr —a/3)/(p-f-r —a—/?).
Note that the above examples can also be applied to the case
min (p, t) ^max (a, P).
Bibliography
1. E. Landau,
Handbuch
der Lehre von der Verteilung
der Primzahlen,
vol. II,
Teubner, Leibzig, 1909.
2. H. Bohr, Bidrag til de Dirichlet'ske Raekkers Theori, Dissertation
for the degree
of Doctor of Philosophy,
Copenhagen,
1910. Both the dissertation
and an English
translation
appear in Bohr's Collected Mathematical
Works, volumes I and III, respectively, Danish Mathematical
Society, Copenhagen, 1952.
Ohio State University
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