Zeta functions for the Riemann zeros

Voros, André

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 665-699.

Certaines fonctions Zêta définies sur les zéros de Riemann, par une famille de séries de Dirichlet, se prolongent à tout le plan complexe en des fonctions méromorphes, dont de nombreuses caractéristiques peuvent être explicitées (la structure polaire, mais aussi une infinité de valeurs spéciales).

A family of Zeta functions built as Dirichlet series over the Riemann zeros are shown to have meromorphic extensions in the whole complex plane, for which numerous analytical features (the polar structures, plus countably many special values) are explicitly displayed.

Classification : 11Mxx, 30B40, 30B50
Keywords: Riemann Zeta function, Riemann zeros, Dirichlet series, Hadamard factorization, meromorphic functions, Mellin transform
Mot clés : fonction Zêta de Riemann, zéros de Riemann, séries de Dirichlet, factorisation de Hadamard, fonctions méromorphes, transformation de Mellin

[1] M. Abramowitz; I.A. Stegun Handbook of Mathematical Functions, chap. 23, Dover, New York, 1965

[2] T.M. Apostol Formulas for Higher Derivatives of the Riemann Zeta Function, Math. Comput, Volume 44 (1985), pp. 223-232 | DOI | MR | Zbl

[3] N.L. Balazs; C. Schmit; A. Voros Spectral Fluctuations and Zeta Functions, J. Stat. Phys, Volume 46 (1987), pp. 1067-1090 | DOI | MR | Zbl

[4] M.V. Berry; J.P. Keating A new asymptotic representation for $\zeta (\frac{1}{2}+\mathrm{i}t)$ and quantum spectral determinants, Proc. R. Soc. Lond, Volume A437 (1992), pp. 151-173 | Zbl

[5] J.M. Borwein; D.M. Bradley; R.E. Crandall Computational strategies for the Riemann Zeta Function, J. Comput. Appl. Math, Volume 121 (2000), pp. 247-296 | DOI | MR | Zbl

[6] P. Cartier; M. Waldschmidt, P. Moussa An Introduction to Zeta Functions, Number Theory to Physics (1992), pp. 1-63 | Zbl

[7] P. Cartier; A. Voros; eds. P. Cartier et al. Une nouvelle interprétation de la formule des traces de Selberg, Progress in Mathematics (C. R. Acad. Sci. Paris, Série I ; The Grothendieck Festschrift), Volume 307 ; vol. 2 (1988 ; 1990), p. 143-148 ; 1--67 | Zbl

[8] I.C. Chakravarty The Secondary Zeta-functions, J. Math. Anal. Appl, Volume 30 (1970), pp. 280-294 | DOI | MR | Zbl

[9] B.K. Choudhury The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond., Volume 450 (1995), pp. 477-499 | DOI | MR | Zbl

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

[11] H. Davenport Multiplicative Number Theory, Graduate Texts in Mathematics, 74, Springer-Verlag, 2000 | MR | Zbl

[12] J. Delsarte Formules de Poisson avec reste, J. Anal. Math. (Jerusalem), Volume 17 (1966) no. (Section 7), pp. 419-431 | DOI | MR | Zbl

[13] C. Deninger Local L-factors of motives and regularized determinants, Invent. Math., Volume 107 (1992) no. Thm 3.3 and Section 4, pp. 135-150 | DOI | MR | Zbl

[14] H.M. Edwards Riemann's Zeta Function, Academic Press, 1974 | Zbl

[15] A. Erdélyi Higher Transcendental Functions (Bateman Manuscript Project), Vols I chap. 1 and III chap. 17, McGraw-Hill, New York, 1953 | Zbl

[16] A. Fujii The zeros of the Riemann zeta function and Gibbs's phenomenon, Comment. Math. Univ. St. Paul. (Japan), Volume 32 (1983), pp. 229-248 | MR | Zbl

[17] A.P. Guinand A summation formula in the theory of prime numbers, Proc. London Math. Soc., Series 2, Volume 50 (1949) no. Section A(A), pp. 107-119 | MR | Zbl

[18] A.P. Guinand Fourier reciprocities and the Riemann zeta-function, Proc. London Math. Soc., Séries 2, Volume 51 (1950), pp. 401-414 | DOI | MR | Zbl

[19] D.A. Hejhal The Selberg trace formula and the Riemann zeta function, Duke Math. J, Volume 43 (1976), pp. 441-481 | DOI | MR | Zbl

[20] G. Illies Regularized products and determinants, Commun. Math. Phys, Volume 220 (2001), pp. 69-94 | DOI | MR | Zbl

[21] P. Jeanquartier Transformation de Mellin et développements asymptotiques, Enseign. Math. II. Ser, Volume 25 (1979), pp. 285-308 | MR | Zbl

[22] J. Jorgenson; S. Lang Basic analysis of regularized series and products, Lecture Notes in Mathematics, 1564, Springer-Verlag, 1993 | MR | Zbl

[23] J. Jorgenson; S. Lang Explicit formulas for regularized products and series, Lecture Notes in Mathematics, 1593, Springer-Verlag, 1994 | MR | Zbl

[24] N. Kurokawa; M.R. Stein and R. Keith Dennis eds. Parabolic components of zeta functions ; Special values of Selberg zeta functions, Algebraic K-theory and algebraic number theory (Proceedings, Honolulu 1987) (Proc. Japan Acad, Ser. A ; Contemp. Math.,), Volume 64 ; 83 (1988 ; 1989), p. 21-24 ; 133--149 | Zbl

[25] P. Lebœuf Prime correlations and their fluctuations, Proceedings of TH-2002 Conference, Paris (Ann. Henri Poincaré), Volume Special Issue (July 2002)

[26] D.H. Lehmer The Sum of Like Powers of the Zeros of the Riemann Zeta Function, Math. Comput, Volume 50 (1988), pp. 265-273 | DOI | MR | Zbl

[27] Yu.V. Matiyasevich A relationship between certain sums over trivial and nontrivial zeros of the Riemann zeta-function, Mat. Zametki ; Translation: Math. Notes (Acad. Sci. USSR), Volume 45 ; 45 (1989 ; 1989), p. 65-70 ; 131--135 | MR | Zbl

[28] Y. Matsuoka On the values of the Riemann zeta function at half integers, Tokyo J. Math, Volume 2 (1979), pp. 371-377 | DOI | MR | Zbl

[29] Y. Matsuoka A note on the relation between generalized Euler constants and the zeros of the Riemann zeta function ; A sequence associated with the zeros of the Riemann zeta function, J. Fac. Educ. Shinshu Univ ; Tsukuba J. Math., Volume 53 ; 10 (1985 ; 1986), p. 81-82 ; 249--254 | Zbl

[30] A.M. Odlyzko The ${10}^{20}$-th zero of the Riemann zeta function and 175 million of its neighbors (1992) (AT & T report, http://www.research.att.com/~amo/)

[31] J.R. Quine; S.H. Heydari; R.Y. Song Zeta regularized products, Trans. Amer. Math. Soc, Volume 338 (1993), pp. 213-231 | DOI | MR | Zbl

[32] B. Randol On the analytic continuation of the Minakshisundaram--Pleijel zeta function for compact Riemann surfaces, Trans. Amer. Math. Soc, Volume 201 (1975), pp. 241-246 | DOI | MR | Zbl

[33] M. Rubinstein; P. Sarnak Chebyshev's bias, Exp. Math, Volume 3 (1994), pp. 173-197 | MR | Zbl

[34] M. Schröter; C. Soulé On a result of Deninger concerning Riemann's zeta function, Motives, Proc. Symp. Pure Math., Part 1, Volume 55 (1994), pp. 745-747 | Zbl

[35] F. Steiner On Selberg's zeta function for compact Riemann surfaces, Phys. Lett., Volume B 188 (1987), pp. 447-454 | MR

[36] E.C. Titchmarsh The Theory of the Riemann Zeta-function, Oxford Univ. Press, 1986 | MR | Zbl

[37] A. Voros Spectral functions, special functions and the Selberg zeta function, Commun. Math. Phys, Volume 110 (1987), pp. 439-465 | MR | Zbl

[38] A. Voros; N. Kurokawa and T. Sunada eds. Spectral zeta functions, Zeta Functions in Geometry (Proceedings, Tokyo 1990) (Advanced Studies in Pure Mathematics ; Math. Soc. Japan), Volume 21 (1990 ; 1992), pp. 327-358 | Zbl

[39] E. Bombieri; J.C. Lagarias Complements to Li's Criterion for the Riemann Hypothesis, J. Number Theory, Volume 77 (1999), pp. 274-287 | DOI | MR | Zbl

[40] A. Voros More zeta functions for the Riemann zeros (June 2003) (Saclay preprint T03/078)

[41] M. Hirano; N. Kurokawa; M. Nakayama Half Zeta functions (2003) (preprint) | MR | Zbl