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

TL;DR

This paper investigates prime number races and demonstrates that certain zero configurations of Dirichlet L-functions off the critical line are necessary to explain observed biases in prime distributions across residue classes.

Contribution

It establishes that the Extended Riemann Hypothesis is essential for understanding prime race biases and links zero configurations off the critical line to the impossibility of certain prime distribution orderings.

Findings

Proves the necessity of off-critical line zeros for prime race biases.

Shows that some orderings of prime counting functions cannot occur under certain zero configurations.

Connects zero distributions of L-functions to prime distribution inequalities.

Abstract

Let $\pi_{q,a}(x)$ denote the number of primes $\le x$ in the progression $a$ modulo $q$. We study subtle inequities in these functions, with $q$ fixed and variable $a$ (sometimes called 'prime race problems'). It is known unconditionally for many triples $(q,a,b)$ that the difference $\pi_{q,a}(x) - \pi_{q,b}(x)$ changes sign infinitely often, although there may be a pronounced bias toward one sign (first observed by Chebyshev in 1853). Similar results for the comparison of three or more prime counting functions all require the assumption of ERH (extended Riemann Hypothesis for the Dirichlet L-functions modulo $q$). In this paper we show that the assumption of ERH is, in a sense, necessary. That is, we prove, for any quadruple $(q,a,b,c)$ with $a,b,c$ co-prime to $q$, that the existence of certain hypothetical configurations of zeros of Dirichlet L-functions lying off the critical line imply that one of the six possible orderings of the three functions $\pi_{q,a}(x), \pi_{q,b}(x), \pi_{q,c}(x)$ does not occur at all for large enough $x$.