Riesz type criteria for $L$-functions in the Selberg class

TL;DR

This paper develops generalized Riesz-type criteria for the Grand Riemann Hypothesis within the Selberg class of $L$-functions, providing new sufficient and necessary conditions without relying on the Ramanujan hypothesis.

Contribution

It introduces a novel criterion for GRH applicable to a broad class of $L$-functions and constructs a subclass with a necessary criterion, extending classical identities to this setting.

Findings

Established a sufficient criterion for GRH for Selberg class $L$-functions.

Constructed a subclass of the Selberg class with a necessary GRH criterion.

Derived new identities and transformation formulas involving Meijer $G$-functions.

Abstract

We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for $L$-functions in this subclass. Identities of Ramanujan-Hardy-Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer $G$-function of the type $G^{n \ 0}_{0 \ n}$.