The Hybrid Euler–Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q{[T]}$

Michael Yiasemides

doi:10.5802/jtnb.1289

The Hybrid Euler–Hadamard Product Formula for Dirichlet

L

-functions in

𝔽_{q} [T]

Michael Yiasemides ¹

¹ School of Mathematical Sciences University of Nottingham Nottingham, NG7 2RD, UK

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 557-619.

Résumé
Abstract

Nous donnons une formule de produit d’Euler–Hadamard hybride pour les fonctions $L$ de Dirichlet du corps $𝔽_{q} [T] .$ Nous déterminons explicitement le terme principal du moment d’ordre $2 k$ du produit eulérien et, en utilisant la théorie des matrices aléatoires, proposons une formule conjecturale pour le moment d’ordre $2 k$ du produit d’Hadamard. Avec une conjecture de scindage, ça nous ramène à une conjecture sur le moment d’ordre $2 k$ des fonctions $L$ de Dirichlet. En faveur de la conjecture de scindage, nous démontrons qu’elle est vraie pour $k = 1, 2 .$ Ce travail est l’analogue pour les corps de fonctions du travail de Bui et Keating. La différence la plus importante est que dans notre cas la formule de produit d’Euler–Hadamard est exacte (sans terme d’erreur).

For Dirichlet $L$ -functions in $𝔽_{q} [T]$ we obtain a hybrid Euler–Hadamard product formula. We explicitly obtain the main term of the $2 k$ -th moment of the Euler product, and we conjecture via random matrix theory the main term of the $2 k$ -th moment of the Hadamard product. Then making a splitting conjecture, this leads to a conjecture for the $2 k$ -th moment of Dirichlet $L$ -functions. Finally, we lend support for the splitting conjecture by proving the cases $k = 1, 2$ . This work is the function field analogue of the work of Bui and Keating, with the most notable difference being that the Euler–Hadamard product formula is exact in this setting (no error term).

Reçu le : 2022-12-06
Accepté le : 2023-10-27
Publié le : 2024-11-13

DOI : 10.5802/jtnb.1289

Classification : 11M06, 11M26, 11M50, 11R59
Mots clés : hybrid Euler–Hadamard product, moments, Dirichlet $L$-functions, function fields, random matrix theory

Affiliations des auteurs :

Michael Yiasemides ¹

¹ School of Mathematical Sciences University of Nottingham Nottingham, NG7 2RD, UK

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {The {Hybrid} {Euler{\textendash}Hadamard} {Product} {Formula} for {Dirichlet} $L$-functions in $\mathbb{F}_q{[T]}$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Michael Yiasemides. The Hybrid Euler–Hadamard Product Formula for Dirichlet $L$-functions in $\mathbb{F}_q{[T]}$. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 557-619. doi : 10.5802/jtnb.1289. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1289/

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