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Operator-valued zeta functions and Fourier analysis

Bender, Carl M and Brody, Dorje C (2019) Operator-valued zeta functions and Fourier analysis Journal of Physics A: Mathematical and Theoretical, 52, 345201. pp. 1-8.

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Abstract

The Riemann zeta function Ϛ(s) is defined as the infinite sum ∑n=1n-s, which converges when Re s ˃ 1. The Riemann hypothesis asserts that the nontrivial zeros of Ϛ(s) lie on the line Re s = ½ . Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex s for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region Re s ˂ 1 by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by Ϛ(s).

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bender, Carl M
Brody, Dorje Cd.brody@surrey.ac.uk
Date : 31 May 2019
DOI : 10.1088/1751-8121/ab25fa
Copyright Disclaimer : © 2019 IOP Publishing Ltd. As the Version of Record of this article is going to be/has been published on a subscription basis, this Accepted Manuscript will be available for reuse under a CC BY-NC-ND 3.0 licence after a 12 month embargo period. Although reasonable endeavours have been taken to obtain all necessary permissions from third parties to include their copyrighted content within this article, their full citation and copyright line may not be present in this Accepted Manuscript version. Before using any content from this article, please refer to the Version of Record on IOPscience once published for full citation and copyright details, as permission may be required. All third party content is fully copyright protected, unless specifically stated otherwise in the figure caption of the Version of Record.
Depositing User : Clive Harris
Date Deposited : 07 Jun 2019 12:33
Last Modified : 14 Aug 2019 12:54
URI: http://epubs.surrey.ac.uk/id/eprint/851953

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