On generalized Li criterion for a certain class of $L-$functions

TL;DR

This paper introduces generalized Li coefficients for a broad class of $L$-functions, establishing a criterion for zero-free regions, providing an arithmetic formula, and performing numerical analysis on shifted Riemann zeta functions.

Contribution

It extends the Li criterion to a wide class of $L$-functions, including automorphic and Rankin-Selberg types, with new formulas and numerical insights.

Findings

Generalized Li criterion proven for the class $ ext{S}^{lat lat}(\sigma_0, \sigma_1)$

Derived an arithmetic formula for $ au$-Li coefficients

Numerical investigation conducted on shifted Riemann zeta functions

Abstract

We define generalized Li coefficients, called $\tau-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$, derive an arithmetic formula for the computation of $\tau-$Li coefficients and conduct numerical investigation of $\tau-$Li coefficients for a certain product of shifts of the Riemann zeta function.