On the argument of zeta-functions of certain cusp forms near the critical line

In this paper, we prove a semi-adelic version of the Kuznetsov formula over number fields. This formula matches a weighted sum made of Fourier coefficients of cusp forms and Eisenstein series with a weighted sum of Kloosterman sums, the latter weight function is a kind of Bessel transform of the former one. We obtain a variant which is valid over all number fields. The admissible weight functions are important in applications, they depend on the archimedean parameters of the representations and show exponential decay. The automorphic vectors are not necessarily spherical in the archimedean aspect. Such formulas are proven to be useful in analytic number theory, e.g., in the estimate of L-functions on the critical line.

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Zeros of quadratic zeta-functions on the critical line

Acta Arithmetica ◽

10.4064/aa-69-1-21-38 ◽

1995 ◽

Vol 69 (1) ◽

pp. 21-38 ◽

Cited By ~ 2

Author(s):

A. Sankaranarayanan

Keyword(s):

Critical Line ◽

Zeta Functions

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A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.51 ◽

2018 ◽

Vol 40 (3) ◽

pp. 612-662

Author(s):

ALEXANDER ADAM ◽

ANKE POHL

Keyword(s):

Transfer Operator ◽

Zeta Functions ◽

Cusp Forms ◽

Dimensional Representation ◽

Triangle Group ◽

Selberg Zeta Function ◽

One Dimensional ◽

Transfer Operators ◽

Finite Dimensional ◽

Automorphic Functions

Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$. The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$. Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$. In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.

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UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

Mathematical Modelling and Analysis ◽

10.3846/mma.2021.12447 ◽

2021 ◽

Vol 26 (1) ◽

pp. 82-93

Author(s):

Aidas Balčiūnas ◽

Violeta Franckevič ◽

Virginija Garbaliauskienė ◽

Renata Macaitienė ◽

Audronė Rimkevičienė

Keyword(s):

Zeta Function ◽

Wide Class ◽

Riemann Zeta Function ◽

Analytic Functions ◽

Pair Correlation ◽

Zeta Functions ◽

Weak Form ◽

Cusp Forms ◽

The Riemann Zeta Function ◽

Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

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