Li coefficients for automorphic L-functions
[Coefficients de Li de fonctions L automorphes]
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 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.

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.

DOI : 10.5802/aif.2311
Classification : 11M26, 11M36, 11S40
Keywords: Automorphic $L$-function, zeta function
Mot clés : fonctions $L$ automorphes, fonction zêta
Lagarias, Jeffrey C. 1

1 University of Michigan Ann Arbor, MI 48109-1043 (USA)
@article{AIF_2007__57_5_1689_0,
     author = {Lagarias, Jeffrey C.},
     title = {Li coefficients for automorphic $L$-functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1689--1740},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {5},
     year = {2007},
     doi = {10.5802/aif.2311},
     mrnumber = {2364147},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2311/}
}
TY  - JOUR
AU  - Lagarias, Jeffrey C.
TI  - Li coefficients for automorphic $L$-functions
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 1689
EP  - 1740
VL  - 57
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2311/
DO  - 10.5802/aif.2311
LA  - en
ID  - AIF_2007__57_5_1689_0
ER  - 
%0 Journal Article
%A Lagarias, Jeffrey C.
%T Li coefficients for automorphic $L$-functions
%J Annales de l'Institut Fourier
%D 2007
%P 1689-1740
%V 57
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2311/
%R 10.5802/aif.2311
%G en
%F AIF_2007__57_5_1689_0
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/

[1] Barnes, E. W. On the expression of Euler’s constant as a definite integral, Messenger of Math., Volume 33 (1903), pp. 59-61

[2] Biane, P.; Pitman, J.; Yor, M. Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc., Volume 38 (2001), pp. 435-465 | DOI | MR | Zbl

[3] Bombieri, E. Remarks on Weil’s quadratic functional in the theory of prime numbers I, Rend. Mat. Acc. Lincei, Ser. IX, Volume 11 (2000), pp. 183-233 | Zbl

[4] Bombieri, E.; Lagarias, J. C. Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory, Volume 77 (1999), pp. 274-287 | DOI | Zbl

[5] Brown, F. C. S. Li’s criterion and zero-free regions of L-functions, J. Number Theory, Volume 111 (2005), pp. 1-32 | DOI | Zbl

[6] Burnol, J.-F. The explicit formula in simple terms (eprint: arxiv math.NT/9810169, v2 22 Nov. 1998)

[7] Burnol, J.-F. Sur les Formules Explicites I : analyse invariante, C. R. Acad. Sci. Paris, Série I, Volume 331 (2000), pp. 423-428 | MR | Zbl

[8] Coffey, M. Relations and positivity results for the derivatives of the Riemann ξ-function, J. Comput. Appl. Math., Volume 166 (2004), pp. 525-534 | DOI | MR | Zbl

[9] Coffey, Mark W. Toward verification of the Riemann hypothesis: application of the Li criterion, Math. Phys. Anal. Geom., Volume 8 (2005) no. 3, pp. 211-255 | DOI | MR | Zbl

[10] Cogdell, J.; Bernstein, J.; Gelbart, S. Analytic theory of L-functions for GL n , An Introduction to the Langlands Program, Birkhäuser, Boston, 2003, pp. 197-228 | MR | Zbl

[11] Cramér, H. Studien über die Nullstellen der Riemannschen Zetafunktion, Math. Zeitschr., Volume 4 (1919), pp. 104-130 | DOI | MR

[12] Davenport, H. Multiplicative Number Theory, Springer Verlag, New York, 2000 (revised and with a preface by H. L. Montgomery) | MR | Zbl

[13] Deninger, C. Local L-factors of motives and regularized determinants, Invent. Math., Volume 107 (1992), pp. 135-150 | DOI | MR | Zbl

[14] Deninger, C. Lefschetz trace formulas and explicit formulas in analytic number theory, J. Reine Angew., Volume 441 (1993), pp. 1-15 | DOI | MR | Zbl

[15] Deninger, C. Evidence for a cohomological approach to analytic number theory, First European Congress of Mathematics, Volume I (1994), pp. 491-510 | MR | Zbl

[16] Deninger, C. Motivic L-functions and regularized determinants, Motives (Proc. Symp. Pure Math.), Volume 55, part I, Amer. Math. Soc., Providence, 1994, pp. 707-743 | MR | Zbl

[17] Deninger, C. Some analogies between number theory and dynamical systems on foliated spaces, Proc. Int. Cong. Math., Volume I (1998), pp. 163-186 | MR | Zbl

[18] Deninger, C. On the nature of the ‘explicit formulas’ in analytic number theory–A simple example, Number Theoretic Methods (Dev. Math.), Volume 8 (2002), pp. 97-118 | Zbl

[19] Deninger, C.; Schröter, M. A distribution-theoretic proof of Guinand’s functional equation for Cramér’s V-function, J. Lond. Math. Soc., Volume 52 (1995), pp. 48-60 | Zbl

[20] Freitas, Pedro A Li-type criterion for zero-free half-planes of Riemann’s zeta function, J. London Math. Soc. (2), Volume 73 (2006) no. 2, pp. 399-414 | DOI | Zbl

[21] Gelbart, S.; Miller, S. D. Riemann’s zeta function and beyond, Bull. Amer. Math. Soc., Volume 41 (2004), pp. 59-112 | DOI | Zbl

[22] Gelfand, I. M.; Kazhdan, D. Representation of the group GL(n,K) where K is a local field, Lie Groups and Their Representations, John Wiley & Sons, New York, 1974, pp. 95-118 | Zbl

[23] Godement, R.; Jacquet, H. Zeta fuctions of simple algebras, Lecture Notes in Math., 260, Springer Verlag, Berlin, 1972 | MR | Zbl

[24] Guinand, A. P. Fourier reciprocities and the Riemann zeta-function, Proc. London Math. Soc., Volume 51 (1949), pp. 401-414 | DOI | MR | Zbl

[25] Haran, S. Riesz potentials and explicit sums in arithmetic, Invent. Math., Volume 101 (1990), pp. 697-703 | DOI | MR | Zbl

[26] Haran, S. Index theory, potential theory and the Riemann hypothesis, L-Functions and Arithmetic (1991), pp. 257-270 | MR | Zbl

[27] Haran, S. The Mysteries of the Real Prime, Oxford Univ. Press, 2001 | MR | Zbl

[28] Ilies, G. Cramér functions and Guinand equations, Acta Arith., Volume 105 (2002), pp. 103-118 | DOI | MR | Zbl

[29] Iwaniec, H.; Kowalski, E. Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl

[30] Iwaniec, H.; Sarnak, P. Perspectives on the analytic theory of L-functions, Geom. Funct. Anal. (2000), pp. 705-741 GAFA 2000 (Tel Aviv 1999) special volume, part II | MR | Zbl

[31] Jacquet, H. Principal L-functions of the linear group, Automorphic Forms, Representations and L-Functions (Proc. Symp. Pure Math.), Volume 33, part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 63-86 | MR | Zbl

[32] Jacquet, H.; Shalika, J. A. On Euler products and the classification of automorphic representations I, Amer. J. Math., Volume 103 (1981), pp. 499-558 | DOI | MR | Zbl

[33] Jorgenson, J.; Lang, S. Guinand’s theorem and functional equations for the Cramér functions, J. Number Theory, Volume 86 (2001), pp. 351-367 | DOI | Zbl

[34] Keiper, J. Power series expansions of Riemann’s ξ-function, Math. Comp., Volume 58 (1992), pp. 765-773 | Zbl

[35] Li, X.-J. The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory, Volume 65 (1997), pp. 325-333 | DOI | MR | Zbl

[36] Li, X.-J. Explicit formulas for Dirichlet and Hecke L-functions, Illinois J. Math, Volume 48 (2004), pp. 491-503 | MR | Zbl

[37] Li, X.-J. An explicit formula for Hecke L-functions (2005) (eprint: arXiv math.NT/0403148 9 Mar. 2004)

[38] Li, Xian-Jin An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, J. Number Theory, Volume 113 (2005) no. 1, pp. 175-200 | DOI | MR | Zbl

[39] Luo, W.-Z.; Rudnick, Z.; Sarnak, P. On the generalized Ramanujan conjecture for GL(n), Automorphic forms, automorphic repesentations and arithmetic (Proc. Symp. Pure Math.), Volume 66, part 2 (1999), pp. 301-310 | MR | Zbl

[40] Maślanka, Krzysztof Li’s criterion for the Riemann hypothesis—numerical approach, Opuscula Math., Volume 24 (2004) no. 1, pp. 103-114 | Zbl

[41] Patterson, S. J. An introduction to the theory of the Riemann zeta function, Cambridge U. Press, 1988 | MR | Zbl

[42] Rudnick, Z.; Sarnak, P. Zeros of principal L-functions and random matrix theory, Duke Math. J., Volume 81 (1996), pp. 269-322 | DOI | MR | Zbl

[43] Voros, A. A sharpening of Li’s criterion for the Riemann hypothesis (eprint: arXiv math.NT/0404213) | Zbl

[44] Voros, A. Spectral zeta functions, Zeta Functions in Geometry (Adv. Studies in Pure Math.), Volume 24, Math. Soc. Japan, 1992, pp. 327-358 | MR | Zbl

[45] Voros, A. Zeta functions for the Riemann zeros, Ann. Inst. Fourier, Volume 53 (2003), pp. 665-699 | DOI | Numdam | MR | Zbl

[46] Weil, A. Sur les ‘formules explicites’ de la théorie des nombres premiers (dédié à M. Riesz), Meddelanden Från Lunds Univ. Mat. Sem. (1952), pp. 252-265 (Also: Œuvres Scientifiques–Collected Papers, Springer Verlag, corrected second printing 1980, Vol. II, p. 48-61.) | Zbl

Cité par Sources :