The prime number race and zeros of Dirichlet $L$-functions off the critical line. III

TL;DR

This paper investigates how hypothetical zeros of Dirichlet L-functions off the critical line could influence the prime number race, showing that such zeros imply a dominant bias in prime counts for certain residue classes.

Contribution

It establishes a link between off-critical-line zeros of Dirichlet L-functions and the dominance of prime counts in specific residue classes, extending understanding of prime distribution biases.

Findings

Zeros off the critical line imply prime count biases.

Prime number race results hold for a set of x with asymptotic density 1.

The work connects L-function zeros to prime distribution phenomena.

Abstract

We show, for any $q\ge 3$ and distinct reduced residues $a,b \pmod q$, the existence of certain hypothetical sets of zeros of Dirichlet $L$-functions lying off the critical line implies that $\pi(x;q,a)<\pi(x;q,b)$ for a set of real $x$ of asymptotic density 1.