Residue classes containing an unexpected number of primes

Abstract

We fix a nonzero integer $a$ and consider arithmetic progressions $amodq$, with $q$ varying over a given range. We show that, for certain specific values of $a$, the arithmetic progressions $amodq$ contain, on average, significantly fewer primes than expected. We improve on results of Fouvry, Bombieri, Friedlander, Iwaniec, Granville, Hildebrandt, and Maier.