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Angles of Gaussian primes

By Zeév Rudnick

Appears in collections : Prime numbers and automatic sequences: determinism and randomness / Nombres premiers et suites automatiques : aléa et déterminisme, Exposés de recherche

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations.

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