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Spectral Zeta Functions Of Graphs And The Riemann Zeta Function In The Critical Strip
Univ Geneva, Sect Math, 2-4 Rue Lievre,Case Postale 64, CH-1211 Geneva 4, Switzerland..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics. Univ Geneva, Sect Math, 2-4 Rue Lievre,Case Postale 64, CH-1211 Geneva 4, Switzerland.
2017 (English)In: Tohoku mathematical journal, ISSN 0040-8735, Vol. 69, no 4, p. 585-610Article in journal (Refereed) Published
Abstract [en]

We initiate the study of spectral zeta functions zeta(X) 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 zeta(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 zeta(s). We relate zeta(Z) to Euler's beta integral and show how to complete it giving the functional equation xi(Z)(1 - s) = xi(Z)(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 zeta(Zd) (s) to the whole plane and identify the poles. From our aymptotics several known special values of zeta(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 F-1 via an Euler-type integral formula due to Picard.

Place, publisher, year, edition, pages
2017. Vol. 69, no 4, p. 585-610
Keywords [en]
Zeta functions, combinatorial Laplacian, functional equations, hypergeometric functions, Riemann Hypothesis
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-344545DOI: 10.2748/tmj/1512183631ISI: 000417277800006OAI: oai:DiVA.org:uu-344545DiVA, id: diva2:1188255
Available from: 2018-03-07 Created: 2018-03-07 Last updated: 2018-03-07Bibliographically approved

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Karlsson, Anders

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