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Discrepancies in the distribution of Gaussian primes

Lucile Devin

TL;DR

The paper develops a general analytic framework for prime-number races that allows infinitely many L-functions to drive the distribution of arithmetic inequalities. It then specializes to Gaussian primes by modeling angular statistics with Hecke L-functions on ℤ[i], proving the existence of limiting logarithmic distributions for suitably chosen test functions and linking their means to central-value vanishing of L-functions. Under assumptions like RH and, conjecturally, nontrivial central-value behavior, the authors derive predictions of biases in the distribution of Gaussian-prime coordinates and twisted angles, supported by numerical evidence. The work intertwines general distribution theory with explicit Hecke-character L-functions to reveal fine-structure biases in Gaussian-prime statistics and to formulate precise conjectures about sign changes and biases. Overall, it provides a rigorous, infinitely parametrized approach to Chebyshev-type biases in a Gaussian-setting, with concrete implications for the distribution of primes Representable as sums of squares and their Gaussian-angle statistics.

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 $1 \bmod 4$.

Discrepancies in the distribution of Gaussian primes

TL;DR

The paper develops a general analytic framework for prime-number races that allows infinitely many L-functions to drive the distribution of arithmetic inequalities. It then specializes to Gaussian primes by modeling angular statistics with Hecke L-functions on ℤ[i], proving the existence of limiting logarithmic distributions for suitably chosen test functions and linking their means to central-value vanishing of L-functions. Under assumptions like RH and, conjecturally, nontrivial central-value behavior, the authors derive predictions of biases in the distribution of Gaussian-prime coordinates and twisted angles, supported by numerical evidence. The work intertwines general distribution theory with explicit Hecke-character L-functions to reveal fine-structure biases in Gaussian-prime statistics and to formulate precise conjectures about sign changes and biases. Overall, it provides a rigorous, infinitely parametrized approach to Chebyshev-type biases in a Gaussian-setting, with concrete implications for the distribution of primes Representable as sums of squares and their Gaussian-angle statistics.

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 -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 .

Paper Structure

This paper contains 15 sections, 17 theorems, 109 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Chebyshev's bias using infinitely many $L$-functions
  3. Motivations and setting
  4. Statement of the main result
  5. Signs changes
  6. Distribution of the angles of Gaussian primes
  7. $L$-functions of Hecke characters on $\mathbf{Z}[i]$
  8. Properties of some Hecke characters and their $L$-function
  9. A character for angles of Gaussian primes
  10. A family of characters for the twisted angles $\tilde{\theta}_p$
  11. Sign of the functional equation and vanishig at the central point
  12. Proofs of Theorem \ref{['Th race Gaussian primes']} and \ref{['Th race Gaussian primes A mod 4']}
  13. Heuristic for the conjectures
  14. Proof of Theorem \ref{['Th_GeneDistLim']}
  15. Vanishing at the central point with PARI/GP

Key Result

Theorem 2.2

Let $\mathcal{S} =\lbrace L(f_m,\cdot) : m\geq 0 \rbrace$ be a sequence of real analytic $L$-functions of degrees $(d_m)_{m\geq 0}$ and analytic conductors $(\mathfrak{q}(f_m))_{m\geq 0}$, and let $\underline{c} = (c_{m})_{m\geq 0}$ be a sequence of real numbers such that the series is convergent. Assume the Riemann Hypothesis is satisfied for all $L(f_m,\cdot)$, $m\geq 0$ and their second momen

Figures (4)

  • Figure 1: $D_1(x)$ for $x \in [2,5\cdot 10^9]$.
  • Figure 2: $D_2(x)$ for $x \in [2,5\cdot 10^9]$.
  • Figure 3: Relative distribution of the angles $\theta_{p}$ for $p\leq 10^8$ : we count the number of angles $\theta_p$ in $200$ subintervals of $[0,\pi]$ and withdraw the mean value; in red equidistribution; in blue the "secondary term" $\frac{\cos x}{\cos 2x} - \frac{1}{2}$.
  • Figure 4: Relative distribution of the angles $\tilde{\theta}_{p}$ for $p\leq 10^8$ : we count the number of angles $\tilde{\theta}_p$ in $200$ subintervals of $[0,\pi]$ and withdraw the mean value; in red equidistribution; in blue the "secondary term" $\frac{1}{\cos x} - \frac{1}{2}$.

Theorems & Definitions (36)

  • Conjecture 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Proposition 2.7
  • Theorem 3.1
  • ...and 26 more