Zeta functions for the Riemann zeros
Annales de l'Institut Fourier, Volume 53 (2003) no. 3, p. 665-699

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.

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).

DOI : https://doi.org/10.5802/aif.1955
Classification:  11Mxx,  30B40,  30B50
Keywords: Riemann Zeta function, Riemann zeros, Dirichlet series, Hadamard factorization, meromorphic functions, Mellin transform 
@article{AIF_2003__53_3_665_0,
     author = {Voros, Andr\'e},
     title = {Zeta functions for the Riemann zeros},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {3},
     year = {2003},
     pages = {665-699},
     doi = {10.5802/aif.1955},
     zbl = {01940707},
     mrnumber = {2008436},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_3_665_0}
}

Voros, André. Zeta functions for the Riemann zeros. Annales de l'Institut Fourier, Volume 53 (2003) no. 3, pp. 665-699. doi : 10.5802/aif.1955. http://www.numdam.org/item/AIF_2003__53_3_665_0/

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

[2] T.M. Apostol Formulas for Higher Derivatives of the Riemann Zeta Function, Math. Comput, Tome 44 (1985), pp. 223-232 | Article | MR 771044 | Zbl 0557.10029

[3] N.L. Balazs; C. Schmit; A. Voros Spectral Fluctuations and Zeta Functions, J. Stat. Phys, Tome 46 (1987), pp. 1067-1090 | Article | MR 893133 | Zbl 0692.58027

[4] M.V. Berry; J.P. Keating A new asymptotic representation for ζ ( 1 2 + i t ) and quantum spectral determinants, Proc. R. Soc. Lond, Tome A437 (1992), pp. 151-173 | Zbl 0776.11048

[5] J.M. Borwein; D.M. Bradley; R.E. Crandall Computational strategies for the Riemann Zeta Function, J. Comput. Appl. Math, Tome 121 (2000), pp. 247-296 | Article | MR 1780051 | Zbl 0972.11077

[6] P. Cartier; M. Waldschmidt, P. Moussa, J.-M. Luck And C. Itzykson Eds. An Introduction to Zeta Functions, Number Theory to Physics, Springer-Verlag (1992), pp. 1-63 | Zbl 0790.11061

[7] P. Cartier; A. Voros; Eds. P. Cartier Et Al. Une nouvelle interprétation de la formule des traces de Selberg, Progress in Mathematics, Birkhäuser (C. R. Acad. Sci. Paris, Série I ; The Grothendieck Festschrift) Tome 307 ; vol. 2 (1988 ; 1990), p. 143-148 ; 1--67 | Zbl 0729.11024

[8] I.C. Chakravarty The Secondary Zeta-functions, J. Math. Anal. Appl, Tome 30 (1970), pp. 280-294 | Article | MR 258767 | Zbl 0201.05202

[9] B.K. Choudhury The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond., Tome 450 (1995), pp. 477-499 | Article | MR 1356175 | Zbl 0835.11033

[10] H. Cramér Studien über die Nullstellen der Riemannschen Zetafunktion, Math. Z., Tome 4 (1919), pp. 104-130 | Article | JFM 47.0289.03 | MR 1544354

[11] H. Davenport Multiplicative Number Theory, Springer-Verlag, Graduate Texts in Mathematics, Tome 74 (2000) | MR 1790423 | Zbl 1002.11001

[12] J. Delsarte Formules de Poisson avec reste, J. Anal. Math. (Jerusalem), Tome 17 (1966) no. (Section 7), pp. 419-431 | Article | MR 218838 | Zbl 0152.11903

[13] C. Deninger Local L-factors of motives and regularized determinants, Invent. Math., Tome 107 (1992) no. Thm 3.3 and Section 4, pp. 135-150 | Article | MR 1135468 | Zbl 0762.14015

[14] H.M. Edwards Riemann's Zeta Function, Academic Press (1974) | Zbl 0315.10035

[15] A. Erdélyi Higher Transcendental Functions (Bateman Manuscript Project), McGraw-Hill, New York Tome Vols I chap. 1 and III chap. 17 (1953) | Zbl 0051.30303

[16] A. Fujii The zeros of the Riemann zeta function and Gibbs's phenomenon, Comment. Math. Univ. St. Paul. (Japan), Tome 32 (1983), pp. 229-248 | MR 713862 | Zbl 0523.10024

[17] A.P. Guinand A summation formula in the theory of prime numbers, Proc. London Math. Soc., Series 2, Tome 50 (1949) no. Section A(A), pp. 107-119 | MR 26086 | Zbl 0031.11003

[18] A.P. Guinand Fourier reciprocities and the Riemann zeta-function, Proc. London Math. Soc., Séries 2, Tome 51 (1950), pp. 401-414 | Article | MR 31513 | Zbl 0039.11503

[19] D.A. Hejhal The Selberg trace formula and the Riemann zeta function, Duke Math. J, Tome 43 (1976), pp. 441-481 | Article | MR 414490 | Zbl 0346.10010

[20] G. Illies Regularized products and determinants, Commun. Math. Phys, Tome 220 (2001), pp. 69-94 | Article | MR 1882400 | Zbl 1025.58012

[21] P. Jeanquartier Transformation de Mellin et développements asymptotiques, Enseign. Math. II. Ser, Tome 25 (1979), pp. 285-308 | MR 570314 | Zbl 0436.46031

[22] J. Jorgenson; S. Lang Basic analysis of regularized series and products, Springer-Verlag, Lecture Notes in Mathematics, Tome 1564 (1993) | MR 1284924 | Zbl 0788.30003

[23] J. Jorgenson; S. Lang Explicit formulas for regularized products and series, Springer-Verlag, Lecture Notes in Mathematics, Tome 1593 (1994) | MR 1329730 | Zbl 0828.11043

[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), Amer. Math. Soc. (Proc. Japan Acad, Ser. A ; Contemp. Math.,) Tome 64 ; 83 (1988 ; 1989), p. 21-24 ; 133--149 | Zbl 0642.10028

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

[26] D.H. Lehmer The Sum of Like Powers of the Zeros of the Riemann Zeta Function, Math. Comput, Tome 50 (1988), pp. 265-273 | Article | MR 917834 | Zbl 0644.10029

[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), Tome 45 ; 45 (1989 ; 1989), p. 65-70 ; 131--135 | MR 1002519 | Zbl 0683.10033

[28] Y. Matsuoka On the values of the Riemann zeta function at half integers, Tokyo J. Math, Tome 2 (1979), pp. 371-377 | Article | MR 560275 | Zbl 0421.10026

[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., Tome 53 ; 10 (1985 ; 1986), p. 81-82 ; 249--254 | Zbl 0623.10027

[30] A.M. Odlyzko The 10 2 0-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, Tome 338 (1993), pp. 213-231 | Article | MR 1100699 | Zbl 0774.30030

[32] B. Randol On the analytic continuation of the Minakshisundaram--Pleijel zeta function for compact Riemann surfaces, Trans. Amer. Math. Soc, Tome 201 (1975), pp. 241-246 | Article | MR 369286 | Zbl 0299.30010

[33] M. Rubinstein; P. Sarnak Chebyshev's bias, Exp. Math, Tome 3 (1994), pp. 173-197 | MR 1329368 | Zbl 0823.11050

[34] M. Schröter; C. Soulé On a result of Deninger concerning Riemann's zeta function, Motives, Proc. Symp. Pure Math., Part 1, Tome 55 (1994), pp. 745-747 | Zbl 0809.11045

[35] F. Steiner On Selberg's zeta function for compact Riemann surfaces, Phys. Lett., Tome B 188 (1987), pp. 447-454 | MR 888533

[36] E.C. Titchmarsh The Theory of the Riemann Zeta-function, Oxford Univ. Press (1986) | MR 882550 | Zbl 0601.10026

[37] A. Voros Spectral functions, special functions and the Selberg zeta function, Commun. Math. Phys, Tome 110 (1987), pp. 439-465 | MR 891947 | Zbl 0631.10025

[38] A. Voros; N. Kurokawa And T. Sunada Eds. Spectral zeta functions, Zeta Functions in Geometry (Proceedings, Tokyo 1990), Kinokuniya, Tokyo (Advanced Studies in Pure Mathematics ; Math. Soc. Japan) Tome 21 (1990 ; 1992), pp. 327-358 | Zbl 0819.11033

[39] E. Bombieri; J.C. Lagarias Complements to Li's Criterion for the Riemann Hypothesis, J. Number Theory, Tome 77 (1999), pp. 274-287 | Article | MR 1702145 | Zbl 0972.11079

[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 1995867 | Zbl 1057.11041