Li coefficients for automorphic $L$-functions

Lagarias, Jeffrey C.

doi:10.5802/aif.2311

Li coefficients for automorphic

L

-functions
[Coefficients de Li de fonctions

L

automorphes]

Lagarias, Jeffrey C.¹

¹ University of Michigan Ann Arbor, MI 48109-1043 (USA)

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1689-1740.

Résumé
Abstract

Xian-Jin Li a montré que l’hypothèse de Riemann est équivalente à la positivité d’une certaine suite de réels $λ_{n}$ $(n = 1, 2, ...)$ . De manière similaire, on associe à une fonction automorphe principale $L (s, π)$ sur $G L (N)$ une suite de réels $λ_{n} (π)$ . On établit une relation entre ces coefficients et les valeurs prises par la fonctionnelle quadratique de Weil associée à la représentation $π$ , sur un espace de fonctions tests convenablement choisi. La positivité de la partie réelle de ces coefficients est équivalente à la conjecture de Riemann pour $L (s, π)$ . En supposant que l’hypothèse de Riemann est satisfaite pour $L (s, π)$ , on montre que : $λ_{n} (π) = \frac{N}{2} (n log n) + C_{1} (π) n + O (\sqrt{n} log n)$ , où $C_{1} (π)$ est une constante réelle. On construit une fonction entière $F_{π} (z)$ , de type exponentielle, qui interpole ces coefficients de Li généralisés en les valeurs entières de la variable. En supposant que l’hypothèse de Riemann est satisfaite pour $L (s, π)$ , la restriction de cette fonction à l’axe réel admet une transformé de Fourier qui est une distribution tempérée, dont le support est un sous-sensemble dénombrable de $[- π, π]$ , ayant le point $0$ comme unique point d’accumulation.

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients $λ_{n}$ $(n = 1, 2, ...)$ . We define similar coefficients $λ_{n} (π)$ associated to principal automorphic $L$ -functions $L (s, π)$ over $G L (N)$ . We relate these cofficients to values of Weil’s quadratic functional associated to the representation $π$ on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for $L (s, π)$ . Assuming the Riemann hypothesis for $L (s, π)$ , we show that $λ_{n} (π) = \frac{N}{2} n log n + C_{1} (π) n + O (\sqrt{n} log n),$ where $C_{1} (π)$ is a real-valued constant. We construct an entire function $F_{π} (z)$ of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for $L (s, π)$ , this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in $[- π, π]$ having $0$ as its only limit point.

DOI : 10.5802/aif.2311

Classification : 11M26, 11M36, 11S40
Keywords: Automorphic $L$-function, zeta function
Mot clés : fonctions $L$ automorphes, fonction zêta

Affiliations des auteurs :

Lagarias, Jeffrey C. ¹

¹ University of Michigan Ann Arbor, MI 48109-1043 (USA)

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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Lagarias, Jeffrey C. Li coefficients for automorphic $L$-functions. Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1689-1740. doi : 10.5802/aif.2311. https://aif.centre-mersenne.org/articles/10.5802/aif.2311/

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