Mean-periodicity and zeta functions

Fesenko, Ivan; Ricotta, Guillaume; Suzuki, Masatoshi

Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1819-1887.

Cet article établit de nouveaux ponts entre les fonctions zeta en théorie des nombres et l’analyse harmonique moderne, c’est-à-dire entre la classe des fonctions de la variable complexe, qui contient les fonctions zeta des schémas arithmétiques et est stable par produit et quotient, et la classe des fonctions moyennes périodiques sur pluieurs espaces de fonctions de la droite réelle. En particulier, il est démontré que le prolongement méromorphe et l’équation fonctionnelle de la fonction zeta d’un schéma arithmétique correspond à la moyenne périodicité d’une fonction explicitement définie et associée à cette fonction zeta. Le cas des courbes elliptiques sur des corps de nombres et leurs modèles réguliers est traité en détails, et de nombreux exemples supplémentaires sont inclus.

This paper establishes new bridges between zeta functions in number theory and modern harmonic analysis, namely between the class of complex functions, which contains the zeta functions of arithmetic schemes and closed with respect to product and quotient, and the class of mean-periodic functions in several spaces of functions on the real line. In particular, the meromorphic continuation and functional equation of the zeta function of an arithmetic scheme with its expected analytic shape is shown to correspond to mean-periodicity of a certain explicitly defined function associated to the zeta function. The case of elliptic curves over number fields and their regular models is treated in more details, and many other examples are included as well.

[1] Berenstein, Carlos A.; Gay, Roger Complex analysis and special topics in harmonic analysis, Springer-Verlag, New York, 1995 | MR | Zbl

[2] Berenstein, Carlos A.; Struppa, Daniele C. Dirichlet series and convolution equations, Publ. Res. Inst. Math. Sci., Volume 24 (1988) no. 5, pp. 783-810 | DOI | MR | Zbl

[3] Berenstein, Carlos A.; Taylor, B. A. Mean-periodic functions, Internat. J. Math. Math. Sci., Volume 3 (1980) no. 2, pp. 199-235 | DOI | MR | Zbl

[4] Bloch, Spencer De Rham cohomology and conductors of curves, Duke Math. J., Volume 54 (1987) no. 2, pp. 295-308 | DOI | MR | Zbl

[5] Cartier, Pierre Mathemagics (a tribute to L. Euler and R. Feynman), Noise, oscillators and algebraic randomness (Chapelle des Bois, 1999) (Lecture Notes in Phys.), Volume 550, Springer, Berlin, 2000, pp. 6-67 | MR | Zbl

[6] Chowla, Sarvadaman; Selberg, Atle On Epstein’s zeta function. I, Proc. Nat. Acad. Sci. U. S. A., Volume 35 (1949), pp. 371-374 | DOI | MR | Zbl

[7] Connes, Alain Trace formula in noncommutative geometry and the zeros of the Riemann zeta function, Selecta Math. (N.S.) (1999) no. 5, pp. 29-106 | MR | Zbl

[8] Conrey, J. Brian The Riemann hypothesis, Notices Amer. Math. Soc., Volume 50 (2003) no. 3, pp. 341-353 | MR | Zbl

[9] Davenport, Harold; Heilbronn, Hans A. On the zeros of certain Dirichlet series I, J. London Math. Soc. (1936) no. 11, pp. 181-185 | DOI | MR | Zbl

[10] Davenport, Harold; Heilbronn, Hans A. On the zeros of certain Dirichlet series II, J. London Math. Soc. (1936) no. 11, pp. 307-312 | DOI | Zbl

[11] Delsarte, Jean Les fonctions moyennes-périodiques, Journal de Math. Pures et Appl., Volume 14 (1935), pp. 403-453 | Numdam | Zbl

[12] Fesenko, Ivan Analysis on arithmetic schemes. I, Doc. Math. (2003) no. Extra Vol., pp. 261-284 (Kazuya Kato’s fiftieth birthday) | MR | Zbl

[13] Fesenko, Ivan Adelic approach to the zeta function of arithmetic schemes in dimension two, Mosc. Math. J., Volume 8 (2008) no. 2, p. 273-317, 399–400 | MR | Zbl

[14] Fesenko, Ivan Analysis on arithmetic schemes. II, J. K-Theory, Volume 5 (2010) no. 3, pp. 437-557 | DOI | MR | Zbl

[15] Gamkrelidze, R.V.; Khenkin, G.M. Several complex variables. V, Encyclopaedia of Mathematical Sciences, 54, Springer-Verlag, Berlin, 1993 Complex analysis in partial differential equations and mathematical physics, A translation of Current problems in mathematics. Fundamental directions. Vol. 54 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989 | MR

[16] Gilbert, John E. On the ideal structure of some algebras of analytic functions, Pacific J. Math., Volume 35 (1970), pp. 625-634 | DOI | MR | Zbl

[17] Gonek, Steven M. On negative moments of the Riemann zeta-function, Mathematika, Volume 36 (1989) no. 1, pp. 71-88 | DOI | MR | Zbl

[18] Hejhal, Dennis A. The Selberg trace formula for $\mathrm{PSL}(2,R)$. Vol. I, Springer-Verlag, Berlin, 1976 (Lecture Notes in Mathematics, Vol. 548) | MR | Zbl

[19] Hejhal, Dennis A. On the distribution of $log|{\zeta }^{\prime }(\frac{1}{2}+it)|$, Number theory, trace formulas and discrete groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 343-370 | MR | Zbl

[20] Iwaniec, Henryk; Kowalski, Emmanuel Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004 | MR | Zbl

[21] Kahane, Jean-Pierre Lectures on mean periodic functions, Tata Inst. Fundamental Res., Bombay, 1959 | Zbl

[22] Kowalski, Emmanuel The large sieve, monodromy, and zeta functions of algebraic curves. II. Independence of the zeros, Int. Math. Res. Not. IMRN (2008), pp. Art. ID rnn 091, 57 | MR | Zbl

[23] Kurokawa, Nobushige Gamma factors and Plancherel measures, Proc. Japan Acad. Ser. A Math. Sci., Volume 68 (1992) no. 9, pp. 256-260 | DOI | MR | Zbl

[24] Lax, Peter D. Translation invariant spaces, Acta Math., Volume 101 (1959), pp. 163-178 | DOI | MR | Zbl

[25] Levin, Boris Ya. Lectures on entire functions, Translations of Mathematical Monographs, 150, American Mathematical Society, Providence, RI, 1996 (In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko) | MR | Zbl

[26] Liu, Qing Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics,, 6, Oxford University Press, Oxford, 2006 (Translated from the French by Reinie Erné) | MR | Zbl

[27] Meyer, Ralf A spectral interpretation for the zeros of the Riemann zeta function, Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005, Universitätsdrucke Göttingen, Göttingen, 2005, pp. 117-137 | MR | Zbl

[28] Meyer, Yves Algebraic numbers and harmonic analysis, North-Holland Publishing Co., Amsterdam, 1972 (North-Holland Mathematical Library, Vol. 2) | MR | Zbl

[29] Michel, Philippe Analytic number theory and families of automorphic $L$-functions, Automorphic forms and applications (IAS/Park City Math. Ser.), Volume 12, Amer. Math. Soc., Providence, RI, 2007, pp. 181-295 | MR | Zbl

[30] Nikolʼskiĭ, Nikolai K. Invariant subspaces in the theory of operators and theory of functions, Journal of Mathematical Sciences, Volume 5 (1976) no. 2, pp. 129-249 | DOI | Zbl

[31] Nikolʼskiĭ, Nikolai K. Elementary description of the methods of localizing ideals, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 170 (1989) no. Issled. Linein. Oper. Teorii Funktsii. 17, p. 207-232, 324–325 | MR | Zbl

[32] Pólya, George; Szegő, Gabor Problems and theorems in analysis. I, Classics in Mathematics, Springer-Verlag, Berlin, 1998 (Series, integral calculus, theory of functions, Translated from the German by Dorothee Aeppli, Reprint of the 1978 English translation) | MR

[33] Roquette, Peter Class field theory in characteristic $p$, its origin and development, Class field theory—its centenary and prospect (Tokyo, 1998) (Adv. Stud. Pure Math.), Volume 30, Math. Soc. Japan, Tokyo, 2001, pp. 549-631 | MR | Zbl

[34] Rubinstein, Michael; Sarnak, Peter Chebyshev’s bias, Experiment. Math. 3 (1994) no. 3, pp. 173-197 | DOI | MR | Zbl

[35] Schwartz, Laurent Théorie générale des fonctions moyenne-périodiques, Ann. of Math. (2), Volume 48 (1947), pp. 857-929 | DOI | MR | Zbl

[36] Selberg, Atle Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.), Volume 20 (1956), pp. 47-87 | MR | Zbl

[37] Serre, Jean-Pierre Zeta and $L$ functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 82-92 | MR | Zbl

[38] Serre, Jean-Pierre Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Œuvres. Vol. II, Springer-Verlag, Berlin, 1986 (1960–1971) | Zbl

[39] Soulé, C. On zeroes of automorphic $L$-functions, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999) (Contemp. Math.), Volume 290, Amer. Math. Soc., Providence, RI, 2001, pp. 167-179 | MR | Zbl

[40] Stark, Harold M. On the zeros of Epstein’s zeta function, Mathematika, Volume 14 (1967), pp. 47-55 | DOI | MR | Zbl

[41] Suzuki, Masatoshi Two dimensional adelic analysis and cuspidal automorphic representations of $GL\left(2\right)$ (prepublication, February 2008 to be published in the Proceedings of the workshop “Multiple Dirichlet Series and Applications to Automorphic Forms”)

[42] Suzuki, Masatoshi Positivity of certain functions associated with analysis on elliptic surface, J. Number Theory, Volume 131 (2011), pp. 1770-1796 | DOI | MR | Zbl

[43] Tate, John T. Fourier analysis in number fields, and Hecke’s zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 305-347 | MR

[44] Titchmarsh, Edward C. The theory of the Riemann zeta-function, The Clarendon Press Oxford University Press, New York, 1986 (Edited and with a preface by D. R. Heath-Brown) | MR | Zbl

[45] Widder, David Vernon The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941 | MR | Zbl

[46] Wiles, Andrew Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 443-551 | DOI | MR | Zbl

[47] Zagier, Don Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic (Bombay, 1979) (Tata Inst. Fund. Res. Studies in Math.), Volume 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 275-301 | MR | Zbl