Xian-Jin Li a montré que l’hypothèse de Riemann est équivalente à la positivité d’une certaine suite de réels . De manière similaire, on associe à une fonction automorphe principale sur une suite de réels . On établit une relation entre ces coefficients et les valeurs prises par la fonctionnelle quadratique de Weil associée à la représentation , 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 . En supposant que l’hypothèse de Riemann est satisfaite pour , on montre que : , où est une constante réelle. On construit une fonction entière , 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 , 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 , ayant le point 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 . We define similar coefficients associated to principal automorphic -functions over . We relate these cofficients to values of Weil’s quadratic functional associated to the representation 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 . Assuming the Riemann hypothesis for , we show that where is a real-valued constant. We construct an entire function of exponential type that interpolates the generalized Li coefficients at integer values. Assuming the Riemann hypothesis for , this function on the real axis has a Fourier transform that is a tempered distribution whose support is a countable set in having as its only limit point.
Classification : 11M26, 11M36, 11S40
Mots clés : fonctions automorphes, fonction zêta
@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'institut Fourier}, volume = {57}, number = {5}, year = {2007}, doi = {10.5802/aif.2311}, mrnumber = {2364147}, zbl = {pre05214656}, language = {en}, url = {www.numdam.org/item/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. http://www.numdam.org/item/AIF_2007__57_5_1689_0/
[1] On the expression of Euler’s constant as a definite integral, Messenger of Math., Volume 33 (1903), pp. 59-61
[2] Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc., Volume 38 (2001), pp. 435-465 | Article | MR 1848256 | Zbl 1040.11061
[3] 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 1008.11034
[4] Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory, Volume 77 (1999), pp. 274-287 | Article | Zbl 0972.11079
[5] Li’s criterion and zero-free regions of -functions, J. Number Theory, Volume 111 (2005), pp. 1-32 | Article | Zbl 02208782
[6] The explicit formula in simple terms (eprint: arxiv math.NT/9810169, v2 22 Nov. 1998)
[7] Sur les Formules Explicites I : analyse invariante, C. R. Acad. Sci. Paris, Série I, Volume 331 (2000), pp. 423-428 | MR 1792480 | Zbl 0992.11064
[8] Relations and positivity results for the derivatives of the Riemann -function, J. Comput. Appl. Math., Volume 166 (2004), pp. 525-534 | Article | MR 2041196 | Zbl 1107.11033
[9] Toward verification of the Riemann hypothesis: application of the Li criterion, Math. Phys. Anal. Geom., Volume 8 (2005) no. 3, pp. 211-255 | Article | MR 2177467 | Zbl 1097.11042
[10] Analytic theory of -functions for , An Introduction to the Langlands Program, Birkhäuser, Boston, 2003, pp. 197-228 | MR 1990380 | Zbl 1111.11303
[11] Studien über die Nullstellen der Riemannschen Zetafunktion, Math. Zeitschr., Volume 4 (1919), pp. 104-130 | Article | MR 1544354
[12] Multiplicative Number Theory, Springer Verlag, New York, 2000 (revised and with a preface by H. L. Montgomery) | MR 1790423 | Zbl 0453.10002
[13] Local -factors of motives and regularized determinants, Invent. Math., Volume 107 (1992), pp. 135-150 | Article | MR 1135468 | Zbl 0762.14015
[14] Lefschetz trace formulas and explicit formulas in analytic number theory, J. Reine Angew., Volume 441 (1993), pp. 1-15 | Article | MR 1228608 | Zbl 0782.11034
[15] Evidence for a cohomological approach to analytic number theory, First European Congress of Mathematics, Volume I (1994), pp. 491-510 | MR 1341834 | Zbl 0838.11002
[16] Motivic -functions and regularized determinants, Motives (Proc. Symp. Pure Math.) Volume 55, part I, Amer. Math. Soc., Providence, 1994, pp. 707-743 | MR 1265547 | Zbl 0816.14010
[17] Some analogies between number theory and dynamical systems on foliated spaces, Proc. Int. Cong. Math., Volume I (1998), pp. 163-186 | MR 1648030 | Zbl 0899.14001
[18] 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 1132.11347
[19] 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 0847.11041
[20] 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 | Article | Zbl 1102.11046
[21] Riemann’s zeta function and beyond, Bull. Amer. Math. Soc., Volume 41 (2004), pp. 59-112 | Article | Zbl 1046.11001
[22] Representation of the group where is a local field, Lie Groups and Their Representations, John Wiley & Sons, New York, 1974, pp. 95-118 | Zbl 0348.22011
[23] Zeta fuctions of simple algebras, Lecture Notes in Math., Volume 260, Springer Verlag, Berlin, 1972 | MR 342495 | Zbl 0244.12011
[24] Fourier reciprocities and the Riemann zeta-function, Proc. London Math. Soc., Volume 51 (1949), pp. 401-414 | Article | MR 31513 | Zbl 0039.11503
[25] Riesz potentials and explicit sums in arithmetic, Invent. Math., Volume 101 (1990), pp. 697-703 | Article | MR 1062801 | Zbl 0788.11055
[26] Index theory, potential theory and the Riemann hypothesis, -Functions and Arithmetic (1991), pp. 257-270 | MR 1110396 | Zbl 0744.11042
[27] The Mysteries of the Real Prime, Oxford Univ. Press, 2001 | MR 1872029 | Zbl 1014.11001
[28] Cramér functions and Guinand equations, Acta Arith., Volume 105 (2002), pp. 103-118 | Article | MR 1932761 | Zbl 1020.11054
[29] Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004 | MR 2061214 | Zbl 1059.11001
[30] Perspectives on the analytic theory of -functions, Geom. Funct. Anal. (2000), pp. 705-741 (GAFA 2000 (Tel Aviv 1999) special volume, part II) | MR 1826269 | Zbl 0996.11036
[31] Principal -functions of the linear group, Automorphic Forms, Representations and -Functions (Proc. Symp. Pure Math.) Volume 33, part 2, Amer. Math. Soc., Providence, RI, 1979, pp. 63-86 | MR 546609 | Zbl 0413.12007
[32] On Euler products and the classification of automorphic representations I, Amer. J. Math., Volume 103 (1981), pp. 499-558 | Article | MR 618323 | Zbl 0473.12008
[33] Guinand’s theorem and functional equations for the Cramér functions, J. Number Theory, Volume 86 (2001), pp. 351-367 | Article | Zbl 0993.11044
[34] Power series expansions of Riemann’s -function, Math. Comp., Volume 58 (1992), pp. 765-773 | Zbl 0767.11039
[35] The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory, Volume 65 (1997), pp. 325-333 | Article | MR 1462847 | Zbl 0884.11036
[36] Explicit formulas for Dirichlet and Hecke -functions, Illinois J. Math, Volume 48 (2004), pp. 491-503 | MR 2085422 | Zbl 1061.11048
[37] An explicit formula for Hecke -functions (2005) (eprint: arXiv math.NT/0403148 9 Mar. 2004)
[38] An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials, J. Number Theory, Volume 113 (2005) no. 1, pp. 175-200 | Article | MR 2141763 | Zbl 02207354
[39] On the generalized Ramanujan conjecture for , Automorphic forms, automorphic repesentations and arithmetic (Proc. Symp. Pure Math.) Volume 66, part 2 (1999), pp. 301-310 | MR 1703764 | Zbl 0965.11023
[40] Li’s criterion for the Riemann hypothesis—numerical approach, Opuscula Math., Volume 24 (2004) no. 1, pp. 103-114 | Zbl 05044698
[41] An introduction to the theory of the Riemann zeta function, Cambridge U. Press, 1988 | MR 933558 | Zbl 0641.10029
[42] Zeros of principal -functions and random matrix theory, Duke Math. J., Volume 81 (1996), pp. 269-322 | Article | MR 1395406 | Zbl 0866.11050
[43] A sharpening of Li’s criterion for the Riemann hypothesis (eprint: arXiv math.NT/0404213) | Zbl 05067009
[44] Spectral zeta functions, Zeta Functions in Geometry (Adv. Studies in Pure Math.) Volume 24, Math. Soc. Japan, 1992, pp. 327-358 | MR 1210795 | Zbl 0819.11033
[45] Zeta functions for the Riemann zeros, Ann. Inst. Fourier, Volume 53 (2003), pp. 665-699 | Article | Numdam | MR 2008436 | Zbl 01940707
[46] 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 0049.03205