Sklar, Abe (1956) Summation formulas associated with a class of Dirichlet series. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:07142011-113939348
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.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||1 January 1956|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Benjamin Perez|
|Deposited On:||14 Jul 2011 20:39|
|Last Modified:||26 Dec 2012 04:37|
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