On primitive Dirichlet characters and the Riemann hypothesis

TL;DR

This paper explores the relationship between primitive Dirichlet characters and the Riemann hypothesis, establishing conditional inequalities and their implications for the truth of the hypothesis.

Contribution

It provides a new conditional inequality involving primitive Dirichlet characters that is equivalent to the Riemann hypothesis, and analyzes its validity under different assumptions.

Findings

If RH is true, the inequality holds for all k.

If RH is false, the inequality holds infinitely often and fails infinitely often.

The result links the distribution of primitive characters to RH.

Abstract

For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-\gamma} \phi(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $\gamma$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $\phi(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold.