Citation
Sklar, Abe (1956) Summation formulas associated with a class of Dirichlet series. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/F6R65E76. https://resolver.caltech.edu/CaltechTHESIS:07142011113939348
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 zetafunction (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 wellknown 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 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1956 
Record Number:  CaltechTHESIS:07142011113939348 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:07142011113939348 
DOI:  10.7907/F6R65E76 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6540 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  14 Jul 2011 20:39 
Last Modified:  09 Nov 2022 19:20 
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