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 ${\pi }_{q,a}(x)$. Here ${\pi }_{q,a}(x)$ is the number of primes $\le x$ in the progression $a\mathrm{mod}q$. 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.

arXiv commentPublished in a hard-to-find conference proceedings in 2003