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<p>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 <i>L</i>-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 squares, the odd square is the square of a positive integer congruent to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1 \bmod 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mspace width="0.277778em" /> <mo>mod</mo> <mspace width="0.277778em" /> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Discrepancies in the distribution of Gaussian primes

  • 期刊论文

  • 发表日期：

    2025年11月20日

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  • Lucile Devin

摘要

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 squares, the odd square is the square of a positive integer congruent to \(1 \bmod 4\) 1 mod 4 .