Equivalent criterion for the grand Riemann hypothesis associated to Maass cusp forms

TL;DR

This paper establishes a new criterion equivalent to the grand Riemann hypothesis for L-functions linked to primitive Maass cusp forms over congruence subgroups, using transformation formulas similar to Ramanujan's identity.

Contribution

It introduces an equivalent criterion for the grand Riemann hypothesis based on transformation formulas for Maass cusp forms, extending classical identities to this setting.

Findings

Derived transformation formulas analogous to Ramanujan's identity for Maass cusp forms

Provided an equivalent criterion for the grand Riemann hypothesis in this context

Extended classical identities to the setting of primitive Maass cusp forms

Abstract

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma_0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated to the primitive Maass cusp form over $\Gamma_0(N)$.