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1.

Friedli, Fabien

et al.

Univ Geneva, Sect Math, 2-4 Rue Lievre,Case Postale 64, CH-1211 Geneva 4, Switzerland..

Karlsson, Anders

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.

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.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

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.

3.

Mazorchuk, Volodymyr

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.

We classify all simple supermodules over the queer Lie superalgebra q _{2} up to classification of equivalence classes of irreducible elements in a certain Euclidean ring.