Generalized Riemann hypotheses: sufficient and equivalent criteria

TL;DR

This paper introduces new criteria that are both sufficient and equivalent for the generalized Riemann Hypothesis, involving zero-free regions, Hadamard-product formulas, and properties of L-functions.

Contribution

It provides novel conditions that deepen understanding of the generalized Riemann Hypothesis and its related properties.

Findings

New sufficient and equivalent criteria for the generalized Riemann Hypothesis

Insights into zero-free regions of L-functions

Analysis of the behavior of L-functions and their ratios

Abstract

This paper presents new sufficient and equivalent conditions for the generalized version of the Riemann Hypothesis. The paper derives also statements and remarks concerning zero-free regions, modified Hadamard-product formulas and the behaviour of $|\frac{L(\bar{\chi},s)}{L(\chi,1-s)}|$.