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

TL;DR

This paper investigates how hypothetical zero configurations of Dirichlet L-functions off the critical line influence the distribution of primes in arithmetic progressions, constructing barriers that control the relative sizes of prime counting functions.

Contribution

It introduces methods to construct barriers affecting the relative magnitudes of prime counting functions based on zeros off the critical line, extending previous analyses.

Findings

Constructed barriers where st (x) dominates or is dominated by others.

Demonstrated control over the orderings of prime counting functions for large x.

Showed that only a small subset of possible orderings occurs asymptotically.

Abstract

We continue our examination the effects of certain hypothetical configurations of zeros of Dirichlet $L$-functions lying off the critical line ("barriers") on the relative magnitude of the functions $\pi_{q,a}(x)$. Here $\pi_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $\pi_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $\pi_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $\pi_{q,a_i}(x)$ ($1\le i\le r$) occur for large $x$; see Theorem 5.1.