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Spectral zeta functions of graphs and the Riemann zeta function in the critical strip
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
University of Geneva, Mathematics department.
2016 (English)In: Tohoku mathematical journal, ISSN 0040-8735Article in journal (Refereed) Accepted
Abstract [en]

We initiate the study of spectral zeta functions ζX for finite and infinite graphs X, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function ζ(s) is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of ζ(s). We relate ζZ to Euler's beta integral and show how to complete it giving the functional equation ξZ(1−s)=ξZ(s). This function appears in the theory of Eisenstein series although presumably with this spectral intepretation unrecognized. In higher dimensions d we provide a meromorphic continuation of ζZd(s) to the whole plane and identify the poles. From our aymptotics several known special values of ζ(s) are derived as well as its non-vanishing on the line Re(s)=1. We determine the spectral zeta functions of regular trees and show it to be equal to a specialization of Appell's hypergeometric function F1 via an Euler-type integral formula due to Picard.

Place, publisher, year, edition, pages
2016.
Keyword [en]
zeta functions, spectral graph theory
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-291433OAI: oai:DiVA.org:uu-291433DiVA: diva2:925525
Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2016-05-11Bibliographically approved

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http://arxiv.org/abs/1410.8010

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Karlsson, Anders
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