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Unformatted text preview: arXiv:math.NT/0112254 v1 22 Dec 2001 On Fourier and Zeta(s) JeanFran cois Burnol Laboratoire Dieudonn e Universit e de Nice Parc Valrose 06108 Nice cedex 02 France burnol@math.unice.fr December 13, 2001 Abstract We study some of the interactions between the Fourier Transform and the Riemann zeta function (and DirichletDedekindHeckeTate Lfunctions). Contents 1 Introduction 2 2 On the Explicit Formulae and log  x  + log  y  3 3 On adeles, ideles, scattering and causality 7 4 On Poisson, Tate, and coPoisson 10 5 On the NymanBeurling criterion, the socalled HilbertP olya idea, and the B aezDuarte, Balazard, Landreau and Saias theorem 18 6 On de Branges Sonine spaces, the spaces HP , and the vectors Z ,k 25 7 On the zeta function, the renormalization group, and duality 32 8 References 36 1 1 Introduction The zeta function ( s ) assumes in Riemanns paper quite a number of distinct identities: it appears as a Dirichlet series, as an Euler product, as an integral transform, as an Hadamard product 1 .. .We retain three such identities and use them as symbolic vertices for a triangle: n 1 n s Q (1 s ) Q p 1 1 p s These formulae stand for various aspects of the zeta function (in the bottom left corner we omit the poles and Gamma factor for easier reading; also, as is known, some care is necessary to make the product converge, for example pairing with 1 is enough.) For the purposes of this manuscript, we may tentatively name these various aspects as follows: summations zeros primes A reading, even casual, of Riemanns paper reveals how much Fourier analysis lies at its heart, on a par with the theory of functions of the complex variable. Let us enhance appropriately the triangle: n 1 n s Fourier Fourier Q (1 s ) Fourier Q p 1 1 p s Indeed, each of the three edges is an arena of interaction between the Fourier Transform, in various incarnations, and the Zeta function (and Dirichlet Lseries, or even more general number theoretical zeta functions.) We specialize to a narrower triangle: n 1 n s summations Hilbert spaces and vectors ? Adeles and Ideles Q (1 s ) zeros Explicit Formulae Q p 1 1 p s primes The big question mark serves as a remainder that we are missing the 2cell (or 2cells) which would presumably be there if the nature of the Riemann zeta function was really understood. 1 Riemann explains how log ( s ) may be written as an infinite sum involving the zeros. 2 It is to be expected that some of the spirit of the wellknown ideas from the theory of function fields will at some point be incorporated into the strengthening of the 1cells, but we shall not discuss this here. These ideas are also perhaps relevant to adding a 2cell, or rather even a 3cell, and we have only contributed to some basic aspects of the downtoearth 1cells....
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This note was uploaded on 09/23/2010 for the course MATH 1121 taught by Professor Dr.mcgrawhill during the Spring '10 term at SUNY Buffalo.
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