Let $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number field of class number $1$ and $\mathcal{O}_\mathbb{K}$ its ring of integers. We study a family of Hecke $L$-functions associated to angular characters on the non-zero ideals of $\mathcal{O}_\mathbb{K}$. Using the powerful Ratios Conjecture (RC) due to Conrey, Farmer, and Zirnbauer, we compute a conditional asymptotic for the average $1$-level density of the zeros of this family, including terms of lower order than the main term in the Katz–Sarnak Density Conjecture coming from random matrix theory. We also prove an unconditional result about the $1$-level density, which agrees with the RC prediction when our test functions have Fourier transforms with support in $(-1,1)$.
Soient $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ un corps quadratique imaginaire de nombre de classes égal à $1$ et $\mathcal{O}_\mathbb{K}$ son anneau des entiers. On étudie une famille de fonctions $L$ de Hecke associées à des caractères angulaires sur les idéaux non nuls de $\mathcal{O}_\mathbb{K}$. En employant la puissante Ratios Conjecture (RC) de Conrey, Farmer et Zirnbauer, on calcule une expression asymptotique conditionnelle pour la densité moyenne de niveau $1$ des zéros de cette famille. Cette estimation comprend des termes d’ordre inférieur au terme principal dans la Density Conjecture de Katz et Sarnak. En outre, on démontre un résultat inconditionnel sur la densité de niveau $1$, qui concorde avec la prédiction de la RC à condition que les fonctions test soient telles que l’intervalle $(-1,1)$ contienne les supports de leurs transformées de Fourier.
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Keywords: $1$-level density, Hecke $L$-functions, the Ratios Conjecture, imaginary quadratic number fields
Kristian Holm 1
CC-BY-ND 4.0
@article{JTNB_2025__37_1_237_0,
author = {Kristian Holm},
title = {The $1${-Level} {Density} for {Zeros} of {Hecke} $L${-Functions} of {Imaginary} {Quadratic} {Number} {Fields} of {Class} {Number} $1$},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {237--283},
year = {2025},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {37},
number = {1},
doi = {10.5802/jtnb.1320},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1320/}
}
TY - JOUR AU - Kristian Holm TI - The $1$-Level Density for Zeros of Hecke $L$-Functions of Imaginary Quadratic Number Fields of Class Number $1$ JO - Journal de théorie des nombres de Bordeaux PY - 2025 SP - 237 EP - 283 VL - 37 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1320/ DO - 10.5802/jtnb.1320 LA - en ID - JTNB_2025__37_1_237_0 ER -
%0 Journal Article %A Kristian Holm %T The $1$-Level Density for Zeros of Hecke $L$-Functions of Imaginary Quadratic Number Fields of Class Number $1$ %J Journal de théorie des nombres de Bordeaux %D 2025 %P 237-283 %V 37 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1320/ %R 10.5802/jtnb.1320 %G en %F JTNB_2025__37_1_237_0
Kristian Holm. The $1$-Level Density for Zeros of Hecke $L$-Functions of Imaginary Quadratic Number Fields of Class Number $1$. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 237-283. doi: 10.5802/jtnb.1320
[1] Transcendental number theory, Cambridge University Press, 1975, x+147 pages | Zbl | DOI | MR
[2] Probability and Measure, Anniversary Edition, John Wiley & Sons, 2012, xvii+624 pages | MR | Zbl
[3] A note on the zeros of zeta and -functions, Math. Z., Volume 281 (2015) no. 1-2, pp. 315-332 | Zbl | DOI | MR
[4] A Refined Conjecture for the Variance of Gaussian Primes across Sectors, Exp. Math., Volume 32 (2023) no. 1, pp. 33-53 | DOI | Zbl
[5] Number theory. Volume II: Analytic and modern tools, Graduate Texts in Mathematics, 240, Springer, 2007, xiii+596 pages | MR | Zbl
[6] Autocorrelation of ratios of -functions, Commun. Number Theory Phys., Volume 2 (2008) no. 3, pp. 593-636 | DOI | MR | Zbl
[7] Applications of the ‐functions ratios conjectures, Proc. Lond. Math. Soc., Volume 94 (2007) no. 3, pp. 594-646 | DOI | MR | Zbl
[8] Long mollifiers of the Riemann zeta-function, Mathematika, Volume 40 (1993) no. 1, pp. 71-87 | DOI | MR | Zbl
[9] Low-lying zeros of quadratic Dirichlet -functions: A transition in the ratios conjecture, Q. J. Math., Volume 69 (2018) no. 4, pp. 1129-1149 | Zbl | MR
[10] Gaussian primes in narrow sectors, Mathematika, Volume 48 (2001) no. 1-2, pp. 119-135 | Zbl | DOI | MR
[11] Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I, Math. Z., Volume 1 (1918) no. 4, pp. 357-376 | DOI | MR | Zbl
[12] Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. II, Math. Z., Volume 6 (1920) no. 1-2, pp. 11-51 | DOI
[13] Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004, xi+615 pages | MR | Zbl
[14] Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications, 45, American Mathematical Society, 1999, xi+419 pages | MR | Zbl
[15] Zeroes of zeta functions and symmetry, Bull. Am. Math. Soc., Volume 36 (1999) no. 1, pp. 1-26 | DOI | MR | Zbl
[16] The pair correlation of zeros of the zeta function, Analytic number theory (Proceedings of Symposia in Pure Mathematics), Volume 24, American Mathematical Society, 1973, pp. 181-193 | DOI | Zbl
[17] Small zeros of quadratic -functions, Bull. Aust. Math. Soc., Volume 47 (1993) no. 2, pp. 307-319 | Zbl | DOI | MR
[18] Angles of Gaussian Primes, Isr. J. Math., Volume 232 (2019) no. 1, pp. 159-199 | Zbl | DOI | MR
[19] Lower order terms for the one-level density of a symplectic family of Hecke -functions, J. Number Theory, Volume 221 (2021), pp. 447-483 | Zbl | DOI | MR
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