The first moment of primes in arithmetic progressions: Beyond the Siegel-Walfisz range

Abstract

We investigate the first moment of primes in progressions $$ \sum_{\substack{q\leq x/N \\ (q,a)=1}} \Big(\psi(x; q, a) - \frac x{\varphi(q)}\Big) $$ as $x, N \to \infty$. We show unconditionally that, when $a=1$, there is a significant bias towards negative values, uniformly for $N\leq {\rm e}^{c\sqrt{\log x}}$. The proof combines recent results of the authors on the first moment and on the error term in the dispersion method. More generally, for $a \in \mathbb Z\setminus\{0\}$ we prove estimates that take into account the potential existence (or inexistence) of Landau-Siegel zeros.