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, Online first, 23 p.

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,π×π ′ ) associée aux représentations automorphes cuspidales π et π ′ de GL m et GL m ′ , 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.

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,π×π ′ ) associated to cuspidal automorphic representations π and π ′ of GL m and GL m ′ , respectively. Our main result implies the first nontrivial prime number theorem for such an L-function (with analytic conductor uniformity) in complete generality.

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Révisé le :
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Première publication :
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) 
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Kaneko, Ikuya; Thorner, Jesse. Highly uniform prime number theorems. Annales de l'Institut Fourier, Online first, 23 p.

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