Discrepancies in the distribution of Gaussian primes

Abstract

Motivated by questions of Fouvry and Rudnick on the distribution of Gaussian primes, we develop a very general setting in which one can study inequities in the distribution of analogues of primes through analytic properties of infinitely many $L$-functions. In particular, we give a heuristic argument for the following claim : for more than half of the prime numbers that can be written as a sum of two square, the odd square is the square of a positive integer congruent to $1\mathrm{mod}4$.

arXiv comment29 pages, 4 figures, new version after referee's suggestions : changes in the presentation of section 4, appendix added to present pari/gp codes to check for the vanishing at the central point

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