An explicit formula for Hecke $L$-functions

TL;DR

This paper provides an explicit formula for a sequence of numbers related to Hecke L-functions, linking the positivity of this sequence to the zeros of Hecke polynomials lying on the critical line, thus contributing to understanding their distribution.

Contribution

It introduces an explicit formula for a sequence connected to Hecke L-functions, offering a new approach to analyze zero distribution on the critical line.

Findings

Positivity of the sequence implies zeros lie on the critical line.

The explicit formula connects number sequences to zero distribution.

Provides a new criterion for the Riemann Hypothesis for Hecke L-functions.

Abstract

In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp forms of weight $k$ for Hecke congruence subgroups, lie on the critical line.