The first moment of primes in arithmetic progressions: beyond the Siegel–Walfisz range

Abstract

We investigate the first moment of primes in progressions ∑q⩽x/N(q,a)=1ψ(x;q,a)−xφ(q)as x,N→∞ . We show unconditionally that, when a=1 , there is a significant bias towards negative values, uniformly for N⩽eclogx . 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∈Z∖{0} we prove estimates that take into account the potential existence (or inexistence) of Landau–Siegel zeros.