Vol. 78, No. 1, 1978

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ISSN: 0030-8730
Mean value theorems for a class of Dirichlet series

Donald Michael Redmond

Vol. 78 (1978), No. 1, 191–231
Abstract

In this paper we are concerned with mean value theorems for the summatory functions of a class of Dirichlet series. This class of Dirichlet series is a class of Dirichlet series satisfying functional equations involving multiple gamma factors. If f(s) = a(n)λns is a Dirichlet series satisfying such a functional equation and E(x) is the associated error term (see (1.2) and (1.4), respectively), then we prove 0-estimates for

∫
x     2
0 |E (y)| dy
(1)

and

 ∑
|a(n)|2,
λn≦x
(2)

in the latter case when λn = n. The results we get for (1) improve known results in some cases. Also the general result (1) is applicable in cases where a similar result of Chandrasekharan and Narasimhan is not.

Mathematical Subject Classification
Primary: 10H25, 10H25
Secondary: 10H08
Milestones
Received: 28 September 1977
Revised: 27 December 1977
Published: 1 September 1978
Authors
Donald Michael Redmond