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Jeffrey C. Lagarias
Li coefficients for automorphic $L$-functions
(Coefficients de Li de fonctions $L$ automorphes)
Annales de l'institut Fourier, 57 no. 5 (2007), p. 1689-1740, doi: 10.5802/aif.2311
Article PDF | Reviews MR 2364147 | Zbl pre05214656
Class. Math.: 11M26, 11M36, 11S40
Keywords: Automorphic $L$-function, zeta function

Résumé - Abstract

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients $\lambda _n$ $(n= 1, 2, \ldots )$. We define similar coefficients $\lambda _n(\pi )$ associated to principal automorphic $L$-functions $L(s, \pi )$ over $GL(N)$. 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 $\lambda _n(\pi ) = \frac{N}{2} n \log n + C_1(\pi ) n + O (\sqrt{n}\log {n}),$ where $C_1(\pi )$ is a real-valued constant. We construct an entire function $F_{\pi }(z)$ 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.

Bibliography

[1] E. W. Barnes, “On the expression of Euler’s constant as a definite integral”, Messenger of Math. 33 (1903), p. 59-61  JFM 34.0334.02
[2] P. Biane, J. Pitman & M. Yor, “Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions”, Bull. Amer. Math. Soc. 38 (2001), p. 435-465 Article |  MR 1848256 |  Zbl 1040.11061
[3] E. Bombieri, “Remarks on Weil’s quadratic functional in the theory of prime numbers I”, Rend. Mat. Acc. Lincei, Ser. IX 11 (2000), p. 183-233  Zbl 1008.11034
[4] E. Bombieri & J. C. Lagarias, “Complements to Li’s criterion for the Riemann hypothesis”, J. Number Theory 77 (1999), p. 274-287 Article |  Zbl 0972.11079
[5] F. C. S. Brown, “Li’s criterion and zero-free regions of $L$-functions”, J. Number Theory 111 (2005), p. 1-32 Article |  Zbl 02208782
[6] J.-F. Burnol, “The explicit formula in simple terms”, eprint: arxiv math.NT/9810169, v2 22 Nov. 1998 arXiv
[7] J.-F. Burnol, “Sur les Formules Explicites I : analyse invariante”, C. R. Acad. Sci. Paris, Série I 331 (2000), p. 423-428  MR 1792480 |  Zbl 0992.11064
[8] M. Coffey, “Relations and positivity results for the derivatives of the Riemann $\xi $-function”, J. Comput. Appl. Math. 166 (2004), p. 525-534 Article |  MR 2041196 |  Zbl 1107.11033
[9] Mark W. Coffey, “Toward verification of the Riemann hypothesis: application of the Li criterion”, Math. Phys. Anal. Geom. 8 (2005) no. 3, p. 211-255 Article |  MR 2177467 |  Zbl 1097.11042
[10] J. Cogdell, Analytic theory of $L$-functions for $GL_n$, in J. Bernstein, S. Gelbart, ed., An Introduction to the Langlands Program, Birkhäuser, 2003, p. 197–228  MR 1990380 |  Zbl 1111.11303
[11] H. Cramér, “Studien über die Nullstellen der Riemannschen Zetafunktion”, Math. Zeitschr. 4 (1919), p. 104-130 Article |  MR 1544354 |  JFM 47.0289.03
[12] H. Davenport, Multiplicative Number Theory, Springer Verlag, New York, 2000, revised and with a preface by H. L. Montgomery  MR 1790423 |  Zbl 0453.10002
[13] C. Deninger, “Local $L$-factors of motives and regularized determinants”, Invent. Math. 107 (1992), p. 135-150 Article |  MR 1135468 |  Zbl 0762.14015
[14] C. Deninger, “Lefschetz trace formulas and explicit formulas in analytic number theory”, J. Reine Angew. 441 (1993), p. 1-15 Article |  MR 1228608 |  Zbl 0782.11034
[15] C. Deninger, Evidence for a cohomological approach to analytic number theory, in First European Congress of Mathematics, Birkhäuser, 1994, p. 491-510  MR 1341834 |  Zbl 0838.11002
[16] C. Deninger, Motivic $L$-functions and regularized determinants, Motives, Proc. Symp. Pure Math. 55, part I, Amer. Math. Soc., 1994, p. 707–743  MR 1265547 |  Zbl 0816.14010
[17] C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces, in Proc. Int. Cong. Math., 1998, p. 163-186  MR 1648030 |  Zbl 0899.14001
[18] C. Deninger, On the nature of the ‘explicit formulas’ in analytic number theory–A simple example, in Number Theoretic Methods, Dev. Math., Kluwer, 2002, p. 97-118  Zbl 1132.11347
[19] C. Deninger & M. Schröter, “A distribution-theoretic proof of Guinand’s functional equation for Cramér’s V-function”, J. Lond. Math. Soc. 52 (1995), p. 48-60  Zbl 0847.11041
[20] Pedro Freitas, “A Li-type criterion for zero-free half-planes of Riemann’s zeta function”, J. London Math. Soc. (2) 73 (2006) no. 2, p. 399-414 Article |  Zbl 1102.11046
[21] S. Gelbart & S. D. Miller, “Riemann’s zeta function and beyond”, Bull. Amer. Math. Soc. 41 (2004), p. 59-112 Article |  Zbl 1046.11001
[22] I. M. Gelfand & D. Kazhdan, Representation of the group $GL(n,K)$ where $K$ is a local field, Lie Groups and Their Representations, John Wiley & Sons, 1974, p. 95–118  Zbl 0348.22011
[23] R. Godement & H. Jacquet, Zeta fuctions of simple algebras, Lecture Notes in Math. 260, Springer Verlag, Berlin, 1972  MR 342495 |  Zbl 0244.12011
[24] A. P. Guinand, “Fourier reciprocities and the Riemann zeta-function”, Proc. London Math. Soc. 51 (1949), p. 401-414 Article |  MR 31513 |  Zbl 0039.11503
[25] S. Haran, “Riesz potentials and explicit sums in arithmetic”, Invent. Math. 101 (1990), p. 697-703 Article |  MR 1062801 |  Zbl 0788.11055
[26] S. Haran, Index theory, potential theory and the Riemann hypothesis, in $L$-Functions and Arithmetic, Cambridge Univ. Press, 1991, p. 257-270  MR 1110396 |  Zbl 0744.11042
[27] S. Haran, The Mysteries of the Real Prime, Oxford Univ. Press, 2001  MR 1872029 |  Zbl 1014.11001
[28] G. Ilies, “Cramér functions and Guinand equations”, Acta Arith. 105 (2002), p. 103-118 Article |  MR 1932761 |  Zbl 1020.11054
[29] H. Iwaniec & E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004  MR 2061214 |  Zbl 1059.11001
[30] H. Iwaniec & P. Sarnak, “Perspectives on the analytic theory of $L$-functions”, Geom. Funct. Anal. (2000), p. 705-741, GAFA 2000 (Tel Aviv 1999) special volume, part II  MR 1826269 |  Zbl 0996.11036
[31] H. Jacquet, Principal $L$-functions of the linear group, Automorphic Forms, Representations and $L$-Functions, Proc. Symp. Pure Math. 33, part 2, Amer. Math. Soc., 1979, p. 63–86  MR 546609 |  Zbl 0413.12007
[32] H. Jacquet & J. A. Shalika, “On Euler products and the classification of automorphic representations I”, Amer. J. Math. 103 (1981), p. 499-558 Article |  MR 618323 |  Zbl 0473.12008
[33] J. Jorgenson & S. Lang, “Guinand’s theorem and functional equations for the Cramér functions”, J. Number Theory 86 (2001), p. 351-367 Article |  Zbl 0993.11044
[34] J. Keiper, “Power series expansions of Riemann’s $\xi $-function”, Math. Comp. 58 (1992), p. 765-773  Zbl 0767.11039
[35] X.-J. Li, “The positivity of a sequence of numbers and the Riemann hypothesis”, J. Number Theory 65 (1997), p. 325-333 Article |  MR 1462847 |  Zbl 0884.11036
[36] X.-J. Li, “Explicit formulas for Dirichlet and Hecke $L$-functions”, Illinois J. Math 48 (2004), p. 491-503 Article |  MR 2085422 |  Zbl 1061.11048
[37] X.-J. Li, “An explicit formula for Hecke $L$-functions”, eprint: arXiv math.NT/0403148 9 Mar. 2004, 2005 arXiv
[38] Xian-Jin Li, “An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials”, J. Number Theory 113 (2005) no. 1, p. 175-200 Article |  MR 2141763 |  Zbl 02207354
[39] W.-Z. Luo, Z. Rudnick & P. Sarnak, On the generalized Ramanujan conjecture for $GL(n)$, in Automorphic forms, automorphic repesentations and arithmetic, Proc. Symp. Pure Math., Amer. Math. Soc., 1999, p. 301-310  MR 1703764 |  Zbl 0965.11023
[40] Krzysztof Maślanka, “Li’s criterion for the Riemann hypothesis—numerical approach”, Opuscula Math. 24 (2004) no. 1, p. 103-114  Zbl 05044698
[41] S. J. Patterson, An introduction to the theory of the Riemann zeta function, Cambridge U. Press, 1988  MR 933558 |  Zbl 0641.10029
[42] Z. Rudnick & P. Sarnak, “Zeros of principal $L$-functions and random matrix theory”, Duke Math. J. 81 (1996), p. 269-322 Article |  MR 1395406 |  Zbl 0866.11050
[43] A. Voros, “A sharpening of Li’s criterion for the Riemann hypothesis”, eprint: arXiv math.NT/0404213 arXiv |  Zbl 05067009
[44] A. Voros, Spectral zeta functions, Zeta Functions in Geometry, Adv. Studies in Pure Math. 24, Math. Soc. Japan, 1992, p. 327–358  MR 1210795 |  Zbl 0819.11033
[45] A. Voros, “Zeta functions for the Riemann zeros”, Ann. Inst. Fourier 53 (2003), p. 665-699 Cedram |  MR 2008436 |  Zbl 01940707
[46] A. Weil, “Sur les ‘formules explicites’ de la théorie des nombres premiers (dédié à M. Riesz)”, Meddelanden Från Lunds Univ. Mat. Sem. (1952), p. 252-265, (Also: Œuvres Scientifiques–Collected Papers, Springer Verlag, corrected second printing 1980, Vol. II, p. 48-61.)  Zbl 0049.03205
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