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.
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.
Keywords: Automorphic $L$-function, zeta function
Mot clés : fonctions $L$ automorphes, fonction zêta
Lagarias, Jeffrey C. 1
@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, Volume 57 (2007) no. 5, pp. 1689-1740. doi : 10.5802/aif.2311. https://aif.centre-mersenne.org/articles/10.5802/aif.2311/
[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 | DOI | MR | Zbl
[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
[4] Complements to Li’s criterion for the Riemann hypothesis, J. Number Theory, Volume 77 (1999), pp. 274-287 | DOI | Zbl
[5] Li’s criterion and zero-free regions of -functions, J. Number Theory, Volume 111 (2005), pp. 1-32 | DOI | Zbl
[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 | Zbl
[8] Relations and positivity results for the derivatives of the Riemann -function, J. Comput. Appl. Math., Volume 166 (2004), pp. 525-534 | DOI | MR | Zbl
[9] 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] Analytic theory of -functions for , An Introduction to the Langlands Program, Birkhäuser, Boston, 2003, pp. 197-228 | MR | Zbl
[11] Studien über die Nullstellen der Riemannschen Zetafunktion, Math. Zeitschr., Volume 4 (1919), pp. 104-130 | DOI | MR
[12] Multiplicative Number Theory, Springer Verlag, New York, 2000 (revised and with a preface by H. L. Montgomery) | MR | Zbl
[13] Local -factors of motives and regularized determinants, Invent. Math., Volume 107 (1992), pp. 135-150 | DOI | MR | Zbl
[14] Lefschetz trace formulas and explicit formulas in analytic number theory, J. Reine Angew., Volume 441 (1993), pp. 1-15 | DOI | MR | Zbl
[15] Evidence for a cohomological approach to analytic number theory, First European Congress of Mathematics, Volume I (1994), pp. 491-510 | MR | Zbl
[16] Motivic -functions and regularized determinants, Motives (Proc. Symp. Pure Math.), Volume 55, part I, Amer. Math. Soc., Providence, 1994, pp. 707-743 | MR | Zbl
[17] Some analogies between number theory and dynamical systems on foliated spaces, Proc. Int. Cong. Math., Volume I (1998), pp. 163-186 | MR | Zbl
[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
[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
[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 | DOI | Zbl
[21] Riemann’s zeta function and beyond, Bull. Amer. Math. Soc., Volume 41 (2004), pp. 59-112 | DOI | Zbl
[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
[23] Zeta fuctions of simple algebras, Lecture Notes in Math., 260, Springer Verlag, Berlin, 1972 | MR | Zbl
[24] Fourier reciprocities and the Riemann zeta-function, Proc. London Math. Soc., Volume 51 (1949), pp. 401-414 | DOI | MR | Zbl
[25] Riesz potentials and explicit sums in arithmetic, Invent. Math., Volume 101 (1990), pp. 697-703 | DOI | MR | Zbl
[26] Index theory, potential theory and the Riemann hypothesis, -Functions and Arithmetic (1991), pp. 257-270 | MR | Zbl
[27] The Mysteries of the Real Prime, Oxford Univ. Press, 2001 | MR | Zbl
[28] Cramér functions and Guinand equations, Acta Arith., Volume 105 (2002), pp. 103-118 | DOI | MR | Zbl
[29] Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004 | MR | Zbl
[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 | Zbl
[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 | Zbl
[32] On Euler products and the classification of automorphic representations I, Amer. J. Math., Volume 103 (1981), pp. 499-558 | DOI | MR | Zbl
[33] Guinand’s theorem and functional equations for the Cramér functions, J. Number Theory, Volume 86 (2001), pp. 351-367 | DOI | Zbl
[34] Power series expansions of Riemann’s -function, Math. Comp., Volume 58 (1992), pp. 765-773 | Zbl
[35] The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Theory, Volume 65 (1997), pp. 325-333 | DOI | MR | Zbl
[36] Explicit formulas for Dirichlet and Hecke -functions, Illinois J. Math, Volume 48 (2004), pp. 491-503 | MR | Zbl
[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 | DOI | MR | Zbl
[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 | Zbl
[40] Li’s criterion for the Riemann hypothesis—numerical approach, Opuscula Math., Volume 24 (2004) no. 1, pp. 103-114 | Zbl
[41] An introduction to the theory of the Riemann zeta function, Cambridge U. Press, 1988 | MR | Zbl
[42] Zeros of principal -functions and random matrix theory, Duke Math. J., Volume 81 (1996), pp. 269-322 | DOI | MR | Zbl
[43] A sharpening of Li’s criterion for the Riemann hypothesis (eprint: arXiv math.NT/0404213) | Zbl
[44] Spectral zeta functions, Zeta Functions in Geometry (Adv. Studies in Pure Math.), Volume 24, Math. Soc. Japan, 1992, pp. 327-358 | MR | Zbl
[45] Zeta functions for the Riemann zeros, Ann. Inst. Fourier, Volume 53 (2003), pp. 665-699 | DOI | Numdam | MR | Zbl
[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
Cited by Sources: