Li coefficients for automorphic L-functions
Annales de l'Institut Fourier, Volume 57 (2007) no. 5, p. 1689-1740
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 GL(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)=N 2nlogn+C 1 (π)n+O(nlogn), 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.
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 GL(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)=N 2(nlogn)+C 1 (π)n+O(nlogn), 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.
DOI : https://doi.org/10.5802/aif.2311
Classification:  11M26,  11M36,  11S40
Keywords: Automorphic L-function, zeta function
@article{AIF_2007__57_5_1689_0,
     author = {Lagarias, Jeffrey C.},
     title = {Li coefficients for automorphic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     pages = {1689-1740},
     doi = {10.5802/aif.2311},
     zbl = {pre05214656},
     mrnumber = {2364147},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2007__57_5_1689_0}
}
Li coefficients for automorphic $L$-functions. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1689-1740. doi : 10.5802/aif.2311. https://aif.centre-mersenne.org/item/AIF_2007__57_5_1689_0/

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