We fix a nonzero integer and consider arithmetic progressions , with varying over a given range. We show that, for certain specific values of , the arithmetic progressions contain, on average, significantly fewer primes than expected. We improve on results of Fouvry, Bombieri, Friedlander, Iwaniec, Granville, Hildebrandt, and Maier.
"Residue classes containing an unexpected number of primes." Duke Math. J. 161 (15) 2923 - 2943, 1 December 2012. https://doi.org/10.1215/00127094-1902268