Explicit zero-free regions and a $\tau$-Li-type criterion

TL;DR

This paper establishes explicit criteria involving $ au$-Li coefficients that determine the existence or non-existence of zeros of certain functions related to the Riemann Hypothesis, providing concrete bounds for zero-free regions.

Contribution

It introduces explicit bounds on $ au$-Li coefficients that connect their signs to the presence or absence of zeros outside specific regions, advancing understanding of zero distributions.

Findings

Explicit bounds $N_1$, $N_2$ for $ au$-Li coefficients related to zero-free regions

Non-negative $ au$-Li coefficients imply zeros are outside a certain region

Negative $ au$-Li coefficients indicate zeros outside a certain region

Abstract

$\tau$-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove that certain values of the $\tau$-Li coefficients lead to existence or non-existence of certain zeros. The first main result gives explicit numbers $N_1$ and $N_2$ such that if all real parts of the $\tau$-Li coefficients are non-negative for all indices between $N_1$ and $N_2$, then the function has non zeros outside a certain region. According to the second result, if some of the real parts of the $\tau$-Li coefficients are negative for some index $n$ between numbers $n_1$ and $n_2$, then there is at least one zero outside a certain region.