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

Abstract

We investigate the first moment of primes in progressions $$\sum _{\begin{array}{c}q\le x/N\\ (q,a)=1\end{array}}(\psi (x;q,a)-\frac{x}{\phi (q)})$$ as $x,N\to \infty$. We show unconditionally that, when $a=1$, there is a significant bias towards negative values, uniformly for $N\le {\mathrm{e}}^{c\sqrt{\log x}}$. 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.

10 pages