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Summation Formulas Associated with a Class of Dirichlet Series


Sklar, Abe (1956) Summation Formulas Associated with a Class of Dirichlet Series. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/F6R6-5E76.


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 Ø(s) = ^∞Σ_(n=1) a(n) λ ^(-s)_n defines a meromorphic function Ø(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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics and Physics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Minor Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Apostol, Tom M.
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1956
Record Number:CaltechTHESIS:07142011-113939348
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6540
Deposited By: Benjamin Perez
Deposited On:14 Jul 2011 20:39
Last Modified:11 Aug 2023 19:26

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