Summation formulas associated with a class of Dirichlet series

Abstract

The Poisson summation formula, which gives, under suitable conditions on f(x), and expression for sums of the form ^(n_2)Σ_(n=n_1) f(n) 1 ≤ n_1 < n_2 ≤ ∞ can be derived from the functional equation for the Riemann zeta-function (s). In this thesis a class of Dirichlet series is defined whose members have properties analogous to those of s(s); in particular, each series in the class, written in the form O(s) = ^∞Σ_(n=1) a(n) λ ^(-s)_n defines a meromorphic function O(s) which satisfies a relation analogous to the functional equation of s(s). From this relation an identity for sum of the form Σ_(^λn^(≤x) a(n) (x - λ_n)^q is derived. This identity in turn leads, in a quite simple fashion, to summation formulas which give expressions for sums of the form ^(n_2)Σ_(n=n_1) a(n) f(λ_n) 1 ≤ n_1 ≤ n_2 The summation formulas thus derived include the Poisson and other well-known summation formulas as special cases and in addition embrace many expressions that are new. The formulas are not only of interest in themselves, but also provide a tool for investigating many problems that arise in analytic number theory.