On The Mean Values of an Entire Function Represented By a Dirichlet Series II

Year 1973, Volume: 22 , 0 - 0, 01.01.1973

Jodh Pal Sıngh

https://doi.org/10.1501/Commua1_0000000636

Abstract

In this note we prove a theorem which gives us Information as to how the functions log I§ (a) and log (a) grow relative to each other as a ^00.
Theorem. Let f(s) be an entire function represented by a Dirichlet series, then
lim
a ^00
log Ig (<’) log J8,k (o) X ■! (1 + k/X) (X 0)
(X = 0)
where X = lim
CT-^OO
log log Ig (a)
a
1. In the usual notation,
00
f(s) = s 1
Sn e®^-”, (s = (j it), 0 A, (n 1)
lim n -*oo
00
is an entire function in the sense that the Dirichlet series represen- ting it, is ahsolutely convergent for ali finite s and possesses
where 0
lim
(7-4-00
log log M(g)
(7
X, 00, and M(ct) have their usual meanings.

Keywords

Mean Values, Entire Function Represented, Dirichlet Series

References

• Ankara Üniversitesi – Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics Dergisi