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Murmurations of Dirichlet characters

Kyu-Hwan Lee, Thomas Oliver, Alexey Pozdnyakov

TL;DR

The paper introduces murmuration densities for two Dirichlet-character families obtained by averaging character values over primes in short intervals. The complex-character density uses odd/even characters normalized by Gauss sums, while the real-character density weights real quadratic characters with a smooth compactly supported function, both relating to oscillatory integrals. A Zubrilina-type universality emerges in the real-character setting, and the densities interpolate the phase transition in the 1-level density for a symplectic family of $L$-functions. The results are proven via prime-number theory, Poisson summation, and smoothing techniques under the Generalized Riemann Hypothesis, with extensions to composite conductors and short intervals. Overall, the work extends murmurations to Dirichlet-character families and ties them to fundamental random-matrix predictions for $L$-function zeros.

Abstract

We calculate murmuration densities for two families of Dirichlet characters. The first family contains complex Dirichlet characters normalized by their Gauss sums. Integrating the first density over a geometric interval yields a murmuration function compatible with experimental observations. The second family contains real Dirichlet characters weighted by a smooth function with compact support. We show that the second density exhibits a universality property analogous to Zubrilina's density for holomorphic newforms, and it interpolates the phase transition in the the $1$-level density for a symplectic family of $L$-functions.

Murmurations of Dirichlet characters

TL;DR

The paper introduces murmuration densities for two Dirichlet-character families obtained by averaging character values over primes in short intervals. The complex-character density uses odd/even characters normalized by Gauss sums, while the real-character density weights real quadratic characters with a smooth compactly supported function, both relating to oscillatory integrals. A Zubrilina-type universality emerges in the real-character setting, and the densities interpolate the phase transition in the 1-level density for a symplectic family of -functions. The results are proven via prime-number theory, Poisson summation, and smoothing techniques under the Generalized Riemann Hypothesis, with extensions to composite conductors and short intervals. Overall, the work extends murmurations to Dirichlet-character families and ties them to fundamental random-matrix predictions for -function zeros.

Abstract

We calculate murmuration densities for two families of Dirichlet characters. The first family contains complex Dirichlet characters normalized by their Gauss sums. Integrating the first density over a geometric interval yields a murmuration function compatible with experimental observations. The second family contains real Dirichlet characters weighted by a smooth function with compact support. We show that the second density exhibits a universality property analogous to Zubrilina's density for holomorphic newforms, and it interpolates the phase transition in the the -level density for a symplectic family of -functions.
Paper Structure (16 sections, 18 theorems, 152 equations, 8 figures)

This paper contains 16 sections, 18 theorems, 152 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Asymptotics of double averages
  4. Lemmas for Theorem \ref{['thm.main']}
  5. Lemmas for Theorem \ref{['thm.main-real']}
  6. Proof of Theorem \ref{['thm.main']}
  7. Proof of Theorem \ref{['thm.main-real']}
  8. Analysis of $R_{\Phi,A}(y,X, \delta)$
  9. Analysis of $M_{\Phi,A}(y,X, \delta)$
  10. $1$-level density
  11. Supplementary results
  12. Including composite conductors in Theorem \ref{['thm.main']}
  13. Preliminaries
  14. Geometric Intervals
  15. Short intervals
  16. ...and 1 more sections

Key Result

Theorem 1.1

Fix $y\in\mathbb{R}_{>0}$. If $c\in\mathbb{R}_{>1}$, then and, assuming the Riemann hypothesis, if $\delta \in (\frac{1}{2},1)$, then

Figures (8)

  • Figure 1: (Top) $P_\pm(y, 2^{10}, 2)$ for $y \in [0,10]$ with $+$ in blue and (the imaginary part of) $-$ in red. (Bottom) $\widetilde{P}_\pm(y, 2002, 0.51)$ for $y \in [0,2]$ with $+$ in blue and (the imaginary part of) $-$ in red. The solid curves (in yellow and green) represent the limits given by Theorem \ref{['thm.main']}. The discontinuity around $y=1$ will be explained in Remark \ref{['rem.discont']}.
  • Figure 2: Let $\Phi_+(x) = \mathbbm{1}_{(1,2)}(x)\exp \left(\tfrac{-1}{1-4(x-1.5)^2} \right),$$\Phi_-(x) = \mathbbm{1}_{(-2,-1)}(x)\exp \left(\tfrac{-1}{1-4(-x-1.5)^2} \right).$We plot $M_{\Phi_\pm}(y, 2^{19}, \frac{2}{3})$ for $y \in [0,2]$ with $+$ in blue (resp. $-$ in red). We also plot the right hand side of equation \ref{['eq.limreal']} in green (resp. orange).
  • Figure 3: Let $\Phi_+(x) = \mathbbm{1}_{(1,2)}(x)$ and $\Phi_-(x) = \mathbbm{1}_{(-2,-1)}(x)$. We plot $M_{\Phi_+}(y, 2^{19}, \frac{2}{3})$ (resp. $M_{\Phi_-}(y, 2^{19}, \frac{2}{3})$) in blue (resp. red). We also plot the right hand side of equation \ref{['eq.limreal']} in green (resp. orange).
  • Figure 4: Plot of $\sum_{N \in [X, 2X)}\sum_{\chi \in \mathcal{Q}_\pm(N)} \chi(p)/\tau(\chi)$, for $X = 2^{17}$ and $2 \leq p < 4X$ with $+$ in blue and (the imaginary part of) $-$ in red.
  • Figure 5: Plot of $\frac{1}{X}\sum_{N \in [X, 2X)}\sum_{\chi \in \mathcal{D}_\pm(N)} \chi(p)/\tau(\chi)$ for $X = 2^{10}$ for primes $p$ such that $2 \leq p \leq 10 X$, with $+$ in blue and (the imaginary part of) $-$ in red.
  • ...and 3 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 32 more