Residue classes containing an unexpected number of primes

Abstract

We fix a non-zero integer $a$ and consider arithmetic progressions $a\mathrm{mod}q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a\mathrm{mod}q$ contain, on average, significantly fewer primes than expected.

18 pages. Added a few remarks, changed the numbering of sections, slightly improved results, and made a few corrections

References in corpus (1)

• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions

Cited by in corpus (5)

• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
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• Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges
• Uniform Titchmarsh divisor problems
• The first moment of primes in arithmetic progressions: Beyond the Siegel-Walfisz range