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

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 $π_{q,a}(x)$. Here $π_{q,a}(x)$ is the number of primes $\le x$ in the progression $a \mod q$. In particular, we construct barriers so that $π_{q,1}(x)$ is simultaneously greater than, or simultaneously less than, each of $k$ functions $π_{q,a_i}(x)$ ($1\le i\le k$). We also construct barriers so that only a small number of the $r!$ possible orderings of functions $π_{q,a_i}(x)$ ($1\le i\le r$) occur for large $x$; see Theorem 5.1.