no. 5
Highly uniform prime number theorems
[Théorèmes des nombres premiers hautement uniformes]
Kaneko, Ikuya1 ; Thorner, Jesse2
1 The Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 (USA)
2 Department of Mathematics, University of Illinois, Urbana, IL 61801 (USA)
Annales de l'Institut Fourier, Tome 75 (2025) no. 5, pp. 1901-1923.

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem depending explicitly on the available zero-free region. The class contains the Rankin–Selberg $L$-function $L(s,\pi \times \pi ^{\prime })$ associated to cuspidal automorphic representations $\pi $ and $\pi ^{\prime }$ of $\mathrm{GL}_{m}$ and $\mathrm{GL}_{m^{\prime }}$, respectively. Our main result implies the first nontrivial prime number theorem for such an $L$-function (with analytic conductor uniformity) in complete generality.

Nous prouvons une version très uniforme du théorème des nombres premiers pour une certaine classe de fonctions $L$. L’étendue de $x$ dépend polynomialement du conducteur analytique, et le terme d’erreur est exprimé en termes d’un problème d’optimisation dépendant explicitement de la région sans zéro disponible. La classe contient la fonction $L$ de Rankin–Selberg $L(s, \pi \times \pi ^{\prime })$ associée aux représentations automorphes cuspidales $\pi $ et $\pi ^{\prime }$ de $\mathrm{GL}_{m}$ et $\mathrm{GL}_{m^{\prime }}$, respectivement. Notre résultat principal implique le premier théorème non trivial des nombres premiers pour une telle fonction $L$ (avec uniformité du conducteur analytique) en toute généralité.

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DOI : 10.5802/aif.3698
Keywords: Prime number theorem, zero-free region, Rankin–Selberg $L$-function
Mots-clés : théorèmes des nombres premiers, région sans zéro, fonction $L$ de Rankin–Selberg

Kaneko, Ikuya 1 ; Thorner, Jesse 2

1 The Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 (USA)
2 Department of Mathematics, University of Illinois, Urbana, IL 61801 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Kaneko, Ikuya; Thorner, Jesse. Highly uniform prime number theorems. Annales de l'Institut Fourier, Tome 75 (2025) no. 5, pp. 1901-1923. doi : 10.5802/aif.3698. https://aif.centre-mersenne.org/articles/10.5802/aif.3698/

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