The meromorphic continuation of Kloosterman-Selberg zeta functions

1990 ◽  
pp. 23-35 ◽  
Author(s):  
J. W. Cogdell ◽  
I. I. Piatetski-Shapiro
Keyword(s):  
Zeta Functions ◽  
Meromorphic Continuation

scholarly journals Meromorphic continuation of Koba-Nielsen string amplitudes

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
M. Bocardo-Gaspar ◽  
Willem Veys ◽  
W. A. Zúñiga-Galindo
Keyword(s):  
Characteristic Zero ◽  
Zeta Functions ◽  
Open String ◽  
Local Fields ◽  
Meromorphic Continuation ◽  
Partition Functions ◽  
Interesting Connection ◽  
Bona Fide ◽  
Local Zeta Functions ◽  
String Amplitudes

Abstract In this article, we establish in a rigorous mathematical way that Koba-Nielsen amplitudes defined on any local field of characteristic zero are bona fide integrals that admit meromorphic continuations in the kinematic parameters. Our approach allows us to study in a uniform way open and closed Koba-Nielsen amplitudes over arbitrary local fields of characteristic zero. In the regularization process we use techniques of local zeta functions and embedded resolution of singularities. As an application we present the regularization of p-adic open string amplitudes with Chan-Paton factors and constant B-field. Finally, all the local zeta functions studied here are partition functions of certain 1D log-Coulomb gases, which shows an interesting connection between Koba-Nielsen amplitudes and statistical mechanics.

scholarly journals Natural boundaries for Euler products of Igusa zeta functions of elliptic curves

2018 ◽  
Vol 14 (08) ◽  
pp. 2317-2331
Author(s):  
Marcus du Sautoy
Keyword(s):  
Elliptic Curves ◽  
Zeta Functions ◽  
Symmetric Power ◽  
Meromorphic Continuation ◽  
Natural Boundary ◽  
Integer Polynomials ◽  
Analytic Behavior ◽  
Euler Products ◽  
Natural Boundaries

We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

SPECTRAL ZETA FUNCTIONS FOR THE QUANTUM RABI MODELS

2016 ◽  
Vol 229 ◽  
pp. 52-98 ◽  
Author(s):  
SHINGO SUGIYAMA
Keyword(s):  
Complex Plane ◽  
Zeta Functions ◽  
Bernoulli Polynomials ◽  
Simple Pole ◽  
Meromorphic Continuation ◽  
Spectral Zeta Functions ◽  
Weyl Law

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

scholarly journals Abelian surfaces over totally real fields are potentially modular

2021 ◽  
Author(s):  
George Boxer ◽  
Frank Calegari ◽  
Toby Gee ◽  
Vincent Pilloni
Keyword(s):  
Functional Equations ◽  
Zeta Functions ◽  
Meromorphic Continuation ◽  
Abelian Surfaces ◽  
Quadratic Extensions ◽  
Genus 2 ◽  
Genus One ◽  
Totally Real ◽  
Totally Real Fields

AbstractWe show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ A over ${\mathbf {Q}}$ Q with $\operatorname{End}_{ {\mathbf {C}}}A={\mathbf {Z}}$ End C A = Z . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

ON EPSTEIN'S ZETA FUNCTION OF HUMBERT FORMS

2008 ◽  
Vol 04 (03) ◽  
pp. 387-401 ◽  
Author(s):  
RENAUD COULANGEON
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Numerical Integration ◽  
Complex Number ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Number Fields ◽  
Positive Definite ◽  
Meromorphic Continuation ◽  
General Criterion

The Epstein ζ function ζ(Γ,s) of a lattice Γ is defined by a series which converges for any complex number s such that ℜ s > n/2, and admits a meromorphic continuation to the complex plane, with a simple pole at s = n/2. The question as to which Γ, for a fixed s, minimizes ζ(Γ,s), has a long history, dating back to Sobolev's work on numerical integration, and subsequent papers by Delone and Ryshkov among others. This was also investigated more recently by Sarnak and Strombergsson. The present paper is concerned with similar questions for positive definite quadratic forms over number fields, also called Humbert forms. We define Epstein zeta functions in that context and study their meromorphic continuation and functional equation, this being known in principle but somewhat hard to find in the literature. Then, we give a general criterion for a Humbert form to be finally ζ extreme, which we apply to a family of examples in the last section.

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