On relations equivalent to the generalized Riemann hypothesis for the Selberg class

TL;DR

This paper establishes new equivalences for the Generalized Riemann Hypothesis (GRH) within the Selberg class, linking it to integral and polynomial expressions of the generalized Li coefficients and Euler-Stieltjes constants, applicable to automorphic L-functions.

Contribution

It introduces novel integral and polynomial characterizations of the GRH for functions in the Selberg class, including automorphic L-functions, expanding the analytical tools for studying the hypothesis.

Findings

GRH equivalent to integral expression of Li coefficients

GRH equivalent to Chebyshev polynomial expressions of Li coefficients

Derived relations involving Euler-Stieltjes constants related to GRH

Abstract

In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $\lambda_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(\lambda_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $GL_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-functions.