Li coefficients for automorphic $L$-functions

Lagarias, Jeffrey C.

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

Xian-Jin Li a montré que l’hypothèse de Riemann est équivalente à la positivité d’une certaine suite de réels ${\lambda }_n$ $(n=1,2,...)$. De manière similaire, on associe à une fonction automorphe principale $L(s,\pi )$ sur $GL\left(N\right)$ une suite de réels ${\lambda }_n\left(\pi \right)$. On établit une relation entre ces coefficients et les valeurs prises par la fonctionnelle quadratique de Weil associée à la représentation $\pi$, 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 $L(s,\pi )$. En supposant que l’hypothèse de Riemann est satisfaite pour $L(s,\pi )$, on montre que : ${\lambda }_n\left(\pi \right)=\frac{N}{2}(nlogn)+C_1\left(\pi \right)n+O(\sqrt{n}logn)$, où $C_1\left(\pi \right)$ est une constante réelle. On construit une fonction entière $F_{\pi }\left(z\right)$, 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 $L(s,\pi )$, 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 $[-\pi ,\pi ]$, 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 ${\lambda }_n$ $(n=1,2,...)$. We define similar coefficients ${\lambda }_n\left(\pi \right)$ associated to principal automorphic $L$-functions $L(s,\pi )$ over $GL\left(N\right)$. We relate these cofficients to values of Weil’s quadratic functional associated to the representation $\pi$ 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 $L(s,\pi )$. Assuming the Riemann hypothesis for $L(s,\pi )$, we show that ${\lambda }_n\left(\pi \right)=\frac{N}{2}nlogn+C_1\left(\pi \right)n+O(\sqrt{n}logn),$ where $C_1\left(\pi \right)$ is a real-valued constant. We construct an entire function $F_{\pi }\left(z\right)$ of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for $L(s,\pi )$, this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in $[-\pi ,\pi ]$ having $0$ as its only limit point.

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

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