DOCSLIB.ORG

Mean-Periodicity and Zeta Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mean-Periodicity and Zeta Functions MEAN-PERIODICITY AND ZETA FUNCTIONS IVAN FESENKO, GUILLAUME RICOTTA, AND MASATOSHI SUZUKI ABSTRACT. This paper establishes new bridges between the class of complex functions, which contains 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 Hasse zeta function of arithmetic scheme with its expected analytic shape is shown to imply the mean-periodicity of a certain explicitly defined function associ- ated to the zeta function. Conversely, the mean-periodicity of this function implies the meromorphic continuation and functional equation of the zeta function. This opens a new road to the study of zeta functions via the theory of mean-periodic functions which is a part of modern harmonic analysis. 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. CONTENTS 1. Introduction and flavour of the results2 1.1. Boundary terms in the classical one-dimensional case2 1.2. Mean-periodicity and analytic properties of Laplace transforms3 1.3. Hasse zeta functions and higher dimensional adelic analysis4 1.4. Boundary terms of Hasse zeta functions and mean-periodicity4 1.5. The case of Hasse zeta functions of models of elliptic curves5 1.6. Other results 6 1.7. New perspectives 6 1.8. Organisation of the paper6 2. Mean-periodic functions7 2.1. Generalities 7 2.2. Quick review on continuous and smooth mean-periodic functions8 2.3. Some relevant spaces with respect to mean-periodicity9 2.4. Mean-periodic functions in these relevant spaces 10 2.5. Mean-periodicity and analytic properties of Laplace transforms I 11 2.6. Mean-periodicity and analytic properties of Laplace transforms II 13 3. Mean-periodicity and analytic properties of boundary terms 15 3.1. The general issue 15 3.2. Mean-periodicity and meromorphic continuation of boundary terms 16 3.3. Mean-periodicity and functional equation of boundary terms 16 3.4. Statement of the result 17 4. On a class of mean-periodic functions arising from number theory 18 4.1. Functions which will supply mean-periodic functions 18 4.2. Mean-periodicity of these functions 19 4.3. On single sign property for these mean-periodic functions 23 5. Zeta functions of arithmetic schemes and mean-periodicity 23 5.1. Background on Hasse zeta functions of schemes 23 Date: Version of December 11, 2009. 1 2 I. FESENKO, G. RICOTTA, AND M. SUZUKI 5.2. Conjectural analytic properties of L-functions of elliptic curves 24 5.3. Hasse–Weil zeta functions of elliptic curves 25 5.4. Hasse zeta functions of models of elliptic curves 29 5.5. Hasse zeta functions of schemes and mean-periodicity 32 6. Other examples of mean-periodic functions 35 6.1. Dedekind zeta functions and mean-periodicity 35 6.2. Cuspidal automorphic forms and mean-periodicity 37 6.3. Eisenstein series and mean-periodicity 37 Appendix A. A useful analytic estimate for L-functions 39 References 42 Acknowledgements– The authors would like to thank A. Borichev for his enlightening lectures on mean- periodic functions given at the University of Nottingham in December 2007. This work was partially supported by ESPRC grant EP/E049109. The first author would like to thank the JSPS for its support. The second author is partially financed by the ANR project "Aspects Arithmétiques des Matrices Aléa- toires et du Chaos Quantique". 1. INTRODUCTION AND FLAVOUR OF THE RESULTS 1.1. Boundary terms in the classical one-dimensional case. The completed zeta function of a num- ber field K is defined on e (s) 1 by < È ³bK(s) : ³ ,K(s)³K (s) Æ 1 where ³K(s) is the Dedekind zeta function of K and ³K, (s) is a finite product of ¡-factors defined in 1 (5.3). It satisfies the integral representation (see [51]) Z s ³bK(s) f (x) x d¹A (x) Æ j j K£ AK£ where f is an appropriately normalized function in the Schwartz–Bruhat space on AK and stands j j for the module on the ideles A£ of K. An application of analytic duality on K AK leads to the K ½ decomposition ¡ ¢ ¡ ¢ ³bK(s) » f ,s » fb,1 s ! (s) Æ Å ¡ Å f where fb is the Fourier transform of f , »(f ,s) is an entire function and Z 1 s dx !f (s) h f (x)x Æ 0 x with Z Z ¡ 1 ¡ 1 ¢¢ h f (x) : f (xγβ) x¡ fb x¡ γβ d¹(¯)d¹(γ). Æ¡ γ A1 /K ¯ @K ¡ 2 K £ 2 £ Every continuous function on AK which vanishes on K£ vanishes on K, and the boundary @K£ of K£ is K \ K£ {0}. The meromorphic continuation and the functional equation for ³bK(s) are equivalent Æ to the meromorphic continuation and the functional equation for !f (s). Let us remark that !f (s) is ¡ t ¢ t the Laplace transform of H (t) : h e¡ thanks to the change of variable x e¡ . The properties of f Æ f Æ the functions h f (x) and H f (t), which are called the boundary terms for obvious reason, are crucial in order to have a better understanding of !f (s). We have ¡ 1 ± ¢¡ 1 ¢ h (x) ¹ A K£ f (0) x¡ fb(0) , f Æ¡ K ¡ ¡ 1 ± ¢¡ t ¢ H (t) ¹ A K£ f (0) e fb(0) f Æ¡ K ¡ MEAN-PERIODICITY AND ZETA FUNCTIONS 3 since @K£ is just the single point 0, with the appropriately normalized measure on the idele class group. As a consequence, ! (s) is a rational function of s invariant with respect to f fb and s f 7! 7! (1 s). Thus, ³bK(s) admits a meromorphic continuation to C and satisfies a functional equation with ¡ respect to s (1 s). 7! ¡ 1.2. Mean-periodicity and analytic properties of Laplace transforms. The previous discussion in the one-dimensional classical case naturally leads to the analytic study of Laplace transforms of specific functions. In particular, the meromorphic continuation and functional equation for Mellin transforms of real-valued functions f on R£ of rapid decay at and polynomial order at 0Å is Å Å1 equivalent to the meromorphic continuation and functional equation of Z 1 Z s dx Å1 st !f (s) : h f (x)x H f (t)e¡ dt Æ 0 x Æ 0 with 1 ¡ 1¢ h (x) f (x) "x¡ f x¡ f Æ ¡ ¡ t ¢ and H (t) h e¡ , where " 1 is the sign of the expected functional equation (see Section3). f Æ f Ƨ The functions h f (x) and H f (t) are called the boundary terms by analogy. The main question is the following one. What property of h f (x) or H f (t) implies the meromorphic continuation and functional 1 equation for !f (s)? One sufficient answer is mean-periodicity . Mean-periodicity is an easy general- ization of periodicity; a function g of a functional space X is X -mean-periodic if the space spanned by its translates is not dense in X . When the Hahn-Banach theorem is available in X , g is X -mean- periodic if and only if g satisfies a convolution equation g ' 0 for some non-trivial element ' of ¤ Æ the dual space X ¤ and a suitable convolution . Such a convolution equation may be thought as a ¤ generalized differential equation. Often2 mean-periodic functions are certain limits of exponential polynomials, and the latter were used already by Euler in his method of solving ordinary differential equations with constant coefficients. The functional spaces X , which are relevant for number theory, are mainly 8 >C (R£) the continuous functions on R£ (Section 2.2), <> Å Å X C 1(R£) the smooth functions on R£ (Section 2.2), £ Æ > Å Å :> R R Cpoly1 ( £) the smooth functions on £ of at most polynomial growth (see (2.4)) Å Å in the multiplicative setting, which is related to h f (x) and 8 >C (R) the continuous functions on R (Section 2.2), <> X C 1(R) the smooth functions on R (Section 2.2), Å Æ > :> Cexp1 (R) the smooth functions on R of at most exponential growth (see (2.3)) in the additive setting, which is related to H f (t). A nice feature is that the spectral synthesis holds in both X and X . The general theory of X -mean-periodic functions shows that if h f (x) is X - Å £ £ £ mean-periodic then !f (s) has a meromorphic continuation given by the Mellin–Carleman transform of h f (x) and satisfies a functional equation. Similarly, if H f (t) is X -mean-periodic then !f (s) has a Å meromorphic continuation given by the Laplace–Carleman transform of H f (t) and satisfies a func- tional equation. See Theorem 3.2 for an accurate statement. Note that in general !f (s) can have a meromorphic continuation to C and can satisfy a functional equation without the functions h f (x) and H f (t) being mean-periodic (see Remark 3.1 for explicit examples). 1The general theory of mean-periodicity is recalled in Section2. 2In this case one says that the spectral synthesis holds in X (see Section 2.1). 4 I. FESENKO, G. RICOTTA, AND M. SUZUKI 1.3. Hasse zeta functions and higher dimensional adelic analysis. For a scheme S of dimension n its Hasse zeta function Y s 1 ³S(s) : (1 k(x) ¡ )¡ Æ x S ¡ j j 2 0 whose Euler factors correspond to all closed points x of S, say x S , with finite residue field of car- 2 0 dinality k(x) , is the most fundamental object in number theory.
Load more

Recommended publications
  • In Search of the Riemann Zeros
    In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes In Search of the Riemann Zeros Strings, Fractal Membranes and Noncommutative Spacetimes Michel L. Lapidus 2000 Mathematics Subject Classification. P rim a ry 1 1 A 4 1 , 1 1 G 20, 1 1 M 06 , 1 1 M 26 , 1 1 M 4 1 , 28 A 8 0, 3 7 N 20, 4 6 L 5 5 , 5 8 B 3 4 , 8 1 T 3 0. F o r a d d itio n a l in fo rm a tio n a n d u p d a tes o n th is b o o k , v isit www.ams.org/bookpages/mbk-51 Library of Congress Cataloging-in-Publication Data Lapidus, Michel L. (Michel Laurent), 1956 In search o f the R iem ann zero s : string s, fractal m em b ranes and no nco m m utativ e spacetim es / Michel L. Lapidus. p. cm . Includes b ib lio g raphical references. IS B N 97 8 -0 -8 2 18 -4 2 2 2 -5 (alk . paper) 1. R iem ann surfaces. 2 . F unctio ns, Z eta. 3 . S tring m o dels. 4 . N um b er theo ry. 5. F ractals. 6. S pace and tim e. 7 . G eo m etry. I. T itle. Q A 3 3 3 .L3 7 2 0 0 7 515.93 dc2 2 2 0 0 7 0 60 8 4 5 Cop ying and rep rinting. Indiv idual readers o f this pub licatio n, and no npro fi t lib raries acting fo r them , are perm itted to m ak e fair use o f the m aterial, such as to co py a chapter fo r use in teaching o r research.
    [Show full text]
  • The Weil Conjectures for Curves
    The Weil Conjectures for Curves Caleb Ji Summer 2021 1 Introduction We will explain Weil’s proof of his famous conjectures for curves. For the Riemann hypothe- sis, we will follow Grothendieck’s argument [1]. The main tools used in these proofs are basic results in algebraic geometry: Riemann-Roch, intersection theory on a surface, and the Hodge index theorem. For references: in Section 2 we used [3], while the rest can be found in Hartshorne [2] V.1 and Appendix C (some of it in the form of exercises). 1.1 Statements of the Weil conjectures We recall the statements. Let X be a smooth projective variety of dimension n over Fq. We define its zeta function by 1 ! X tr Z(X; t) := exp N ; r r r=1 where Nr is the number of closed points of X where considered over Fqr . Theorem 1.1 (Weil conjectures). Use the above notation. 1. (Rationality) Z(X; t) is a rational function of t. 2. (Functional equation) Let E be the Euler characteristic of X considered over C. Then 1 Z = ±qnE=2tEZ(t): qnt 3. (Riemann hypothesis) We can write P (t) ··· P (t) Z(t) = 1 2n−1 P0(t) ··· P2n(t) n where P0(t) = 1 − t; P2n(t) = 1 − q t and all the Pi(t) are integer polynomials that can be written as Y Pi(t) = (1 − αijt): j i=2 Finally, jαijj = q . 4. (Betti numbers) The degree of the polynomials Pi are the Betti numbers of X considered over C. 1 Caleb Ji The Weil Conjectures for Curves Summer 2021 d P1 r Note that dt log Z(X; t) = r=0 Nr+1t .
    [Show full text]
  • Counting Points and Acquiring Flesh
    Innovations in Incidence Geometry Volume 00 (XXXX), Pages 000–000 ISSN 1781-6475 Counting points and acquiring flesh Koen Thas Abstract This set of notes is based on a lecture I gave at “50 years of Finite Ge- ometry — A conference on the occasion of Jef Thas’s 70th birthday,” in November 2014. It consists essentially of three parts: in a first part, I in- troduce some ideas which are based in the combinatorial theory underlying F1, the field with one element. In a second part, I describe, in a nutshell, the fundamental scheme theory over F1 which was designed by Deitmar. The last part focuses on zeta functions of Deitmar schemes, and also presents more recent work done in this area. Keywords: Field with one element, Deitmar scheme, loose graph, zeta function, Weyl geometry MSC 2000: 11G25, 11D40, 14A15, 14G15 Contents 1 Introduction 2 2 Combinatorial theory 5 3 Deninger-Manin theory 8 4 Deitmar schemes 10 5 Acquiring flesh (1) 14 arXiv:1508.03997v1 [math.AG] 17 Aug 2015 6 Kurokawa theory 15 7 Graphs and zeta functions 18 2 Thas 8 Acquiring flesh (2) — The Weyl functor depicted 25 1 Introduction For a class of incidence geometries which are defined (for instance coordina- tized) over fields, it often makes sense to consider the “limit” of these geome- tries when the number of field elements tends to 1. As such, one ends up with a guise of a “field with one element, F1” through taking limits of geometries. A general reference for F1 is the recent monograph [21].
    [Show full text]
  • Arxiv:1604.01256V4 [Math.NT] 14 Aug 2017 As P Tends to Infinity, We Are Happy to Ignore Such Primes, Which Are Necessarily finite in Number
    SATO-TATE DISTRIBUTIONS ANDREW V.SUTHERLAND ABSTRACT. In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures≤ presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry. 1. AN INTRODUCTION TO SATO-TATE DISTRIBUTIONS Before discussing the Sato-Tate conjecture and Sato-Tate distributions for abelian varieties, we first consider the more familiar setting of Artin motives, which can be viewed as varieties of dimension zero. 1.1. A first example. Let f Z[x] be a squarefree polynomial of degree d; for example, we may take 3 2 f (x) = x x + 1. Since f has integer coefficients, we can reduce them modulo any prime p to obtain − a polynomial fp with coefficients in the finite field Z=pZ Fp. For each prime p define ' Nf (p) := # x Fp : fp(x) = 0 , f 2 g which we note is an integer between 0 and d. We would like to understand how Nf (p) varies with p. 3 The table below shows the values of Nf (p) when f (x) = x x + 1 for primes p < 60: − p : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Nf (p) 00111011200101013 There does not appear to be any obvious pattern (and we should know not to expect one, the Galois group lurking behind the scenes is nonabelian).
    [Show full text]
  • Arithmetic Zeta-Function
    Arithmetic Zeta-Function Gaurish Korpal1 [email protected] Summer Internship Project Report 14th year Int. MSc. Student, National Institute of Science Education and Research, Jatni (Bhubaneswar, Odisha) Certificate Certified that the summer internship project report \Arithmetic Zeta-Function" is the bona fide work of \Gaurish Korpal", 4th year Int. MSc. student at National Institute of Science Ed- ucation and Research, Jatni (Bhubaneswar, Odisha), carried out under my supervision during June 4, 2018 to July 4, 2018. Place: Mumbai Date: July 4, 2018 Prof. C. S. Rajan Supervisor Professor, Tata Institute of Fundamental Research, Colaba, Mumbai 400005 Abstract We will give an outline of the motivation behind the Weil conjectures, and discuss their application for counting points on projective smooth curves over finite fields. Acknowledgements Foremost, I would like to express my sincere gratitude to my advisor Prof. C. S. Rajan for his motivation. I am also thankful to Sridhar Venkatesh1, Rahul Kanekar 2 and Monalisa Dutta3 for the enlightening discussions. Last but not the least, I would like to thank { Donald Knuth for TEX { Michael Spivak for AMS-TEX { Sebastian Rahtz for TEX Live { Leslie Lamport for LATEX { American Mathematical Society for AMS-LATEX { H`anTh^e´ Th`anhfor pdfTEX { Heiko Oberdiek for hyperref package { Steven B. Segletes for stackengine package { David Carlisle for graphicx package { Javier Bezos for enumitem package { Hideo Umeki for geometry package { Peter R. Wilson & Will Robertson for epigraph package { Jeremy Gibbons, Taco Hoekwater and Alan Jeffrey for stmaryrd package { Lars Madsen for mathtools package { Philipp Khl & Daniel Kirsch for Detexify (a tool for searching LATEX symbols) { TeX.StackExchange community for helping me out with LATEX related problems 1M.Sc.
    [Show full text]
  • The Sound of Fractal Strings and the Riemann Hypothesis
    The Sound of Fractal Strings and the Riemann Hypothesis Michel L. Lapidus Abstract We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly dis- cuss closely related developments, including the theory of (fractal) complex dimen- sions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his col- laborators). Key words: Riemann zeta function, Riemann hypothesis (RH), quantization, quan- tized number theory,fractal strings, geometry and spectra, direct and inverse spectral problems for fractal strings, Minkowski dimension, Minkowski measurability, com- plex dimensions, Weyl–Berry conjecture, fractal drums, infinitesimal shift, spec- tral operator, invertbility, quantized Dirichlet series and Euler product, universality, phase transitions, symmetric and asymmetric criteria for RH. arXiv:1505.01548v1 [math-ph] 7 May 2015 Suggested short title for the paper: ”Fractal Strings and the Riemann Hypothesis.” University of California Department of Mathematics 900 University Avenue Riverside, CA 92521–0135, USA e-mail: [email protected] 1 Contents The Sound of Fractal Strings and the Riemann Hypothesis............. 1 Michel L. Lapidus 1 Riemann Zeros and Spectra of Fractal Strings: An Informal Introduction....................................... ....... 4 2 FractalStringsandMinkowskiDimension ............... ..... 7 3 Characterization of Minkowski Measurability and Nondegeneracy 9 4 TheWeyl–BerryConjectureforFractalDrums............ ..... 13 4.1 Weyl’s asymptotic formula with sharp error term for fractaldrums ....................................
    [Show full text]
  • The Sound of Fractal Strings and the Riemann Hypothesis
    The Sound of Fractal Strings and the Riemann Hypothesis Michel L. LAPIDUS Institut des Hautes Etudes´ Scientifiques 35, route de Chartres 91440 – Bures-sur-Yvette (France) Juillet 2015 IHES/M/15/11 The Sound of Fractal Strings and the Riemann Hypothesis Michel L. Lapidus Abstract We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the author with Carl Pomerance and Helmut Maier, respectively). We also briefly dis- cuss closely related developments, including the theory of (fractal) complex dimen- sions (by the author and many of his collaborators, including especially Machiel van Frankenhuijsen), quantized number theory and the spectral operator (jointly with Hafedh Herichi), and some other works of the author (and several of his col- laborators). Key words: Riemann zeta function, Riemann hypothesis (RH), quantization, quan- tized number theory, fractal strings, geometry and spectra, direct and inverse spectral problems for fractal strings, Minkowski dimension, Minkowski measurability, com- plex dimensions, Weyl–Berry conjecture, fractal drums, infinitesimal shift, spec- tral operator, invertbility, quantized Dirichlet series and Euler product, universality, phase transitions, symmetric and asymmetric criteria for RH. Suggested short title for the paper: ”Fractal Strings and the Riemann Hypothesis.” University of California Department of Mathematics 900 University Avenue Riverside, CA 92521–0135, USA e-mail: [email protected] 1 Contents The Sound of Fractal Strings and the Riemann Hypothesis............. 1 Michel L. Lapidus 1 Riemann Zeros and Spectra of Fractal Strings: An Informal Introduction .
    [Show full text]
  • Fractal Complex Dimensions, Riemann Hypothesis And
    Contemporary Mathematics Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator Hafedh Herichi and Michel L. Lapidus Abstract. A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. The inverse spectral problem which they studied is related to answering the question “Can one hear the shape of a fractal drum?”and was shown in [LaMa2] to have a positive answer for fractal strings 1 whose dimension is c ∈ (0, 1) − { 2 } if and only if the Riemann hypothesis is true. Later on, the spectral operator was introduced semi-heuristically by M. L. Lapidus and M. van Frankenhuijsen in their development of the theory of fractal strings and their complex dimensions [La-vF2, La-vF3] as a map that sends the geometry of a fractal string onto its spectrum. In this survey article, we focus on presenting the results obtained by the authors in [HerLa1] about the invertibility (in a suitable sense) of the spectral operator, which turns out to be intimately related to the critical zeroes of the Riemann zeta function. More specifically, given any c ≥ 0, we show that the spectral operator a = ac, now precisely defined as an unbounded normal operator acting in an appropriate weighted Hilbert space Hc, is ‘quasi-invertible’ (i.e., its truncations are invertible) if and only if the Riemann zeta function ζ = ζ(s) does not have any zeroes on the vertical line Re(s) = c. It follows, in particular, that the associated inverse spectral problem has a positive answer for all possible 1 dimensions c ∈ (0, 1), other than the mid-fractal case when c = 2 , if and only if the Riemann hypothesis is true.
    [Show full text]
  • ZETAS 2018 : Zeta Functions, Polyzeta Functions, Arithmetical Series : Applications to Motives and Number Theory
    WELCOME TO ZETAS 2018 : Zeta Functions, Polyzeta Functions, Arithmetical Series : Applications to Motives and Number Theory LAMA Summer School U. Savoie Mont Blanc - CNRS 18-29 June 2018 LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 1 / 20 LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 2 / 20 Why such a Summer School ? Origin : every two years a Series of International Meetings on Zeta Functions at the CNRS UMI Poncelet Laboratory in Moscow, traditionally in December : 1st (2006) : ZETA 1 2nd (2008) : ZETA 2 . 6th (2016) : ZETA 6 7th (2018) : L-functions and algebraic varieties , in Memory of Alexey Zykin, org. Sergey Gorchinskiy, Marc Hindry, Philippe Lebacque, Sergey Rybakov, Michael Tsfasman. idea : launched by Michel Balazard and Michael Tsfasman to allow communication in the collectivity of the researchers working with zeta functions and L functions, all of them, to gather this collectivity. LAMA Summer School (U. Savoie Mont Blanc - CNRS) 18-29 June 2018 3 / 20 A List of Zeta Functions (WiKipedia...) List of zeta functions - Wikipedia https://en.wikipedia.org/wiki/List_of_zeta_functions List of zeta functions In mathematics, a zeta function is (usually) a function analogous to the original example: the Riemann zeta function Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta function Arithmetic zeta function Artin–Mazur zeta-function of a dynamical system Barnes zeta function or Double zeta function Beurling zeta function of Beurling generalized primes Dedekind zeta-function of a number field Epstein zeta-function of a quadratic form.
    [Show full text]
  • The Analytic Class Number Formula for 1-Dimensional Affine Schemes
    THE ANALYTIC CLASS NUMBER FORMULA FOR 1-DIMENSIONAL AFFINE SCHEMES BRUCE W. JORDAN AND BJORN POONEN Abstract. We derive an analytic class number formula valid for an order in a product of S-integers in global fields, or equivalently for reduced finite-type affine schemes of pure dimension 1 over Z. 1. Introduction Let K be a finite extension of Q. Let OK be its ring of integers. Let ζK (s) be the Dedekind zeta function of K, which is the zeta function of Spec OK . Dedekind [Dir94, Supplement XI, x184, IV], generalizing work of Dirichlet, proved the analytic class number formula, which expresses the residue of ζK (s) at s = 1 in terms of arithmetic invariants (see also Hilbert's Zahlbericht [Hil97, Theorem 56]). More precisely, he proved that 2r1 (2π)r2 hR (1) lim(s − 1)ζK (s) = ; s!1 w jDisc Kj1=2 where r1 is the number of real places, r2 is the number of complex places, h is the class number, R is the unit regulator, w is the number of roots of unity, and Disc K is the discriminant. F. K. Schmidt [Sch31, Satz 21] proved an analogue for a global function field. This could be generalized in several ways: • Replace OK by a non-maximal order. • Replace OK by a ring of S-integers for some finite set S of places of K. • Allow Z-algebras of Krull dimension 1 that are not necessarily integral domains. We will generalize simultaneously in all of these directions, by proving a version of (1) for an order O in a product of S-integers in global fields, or equivalently for a reduced affine finite-type Z-scheme of pure dimension 1 (for the equivalence, see Proposition 4.2).
    [Show full text]
  • Deitmar Schemes, Graphs and Zeta Functions
    DEITMAR SCHEMES, GRAPHS AND ZETA FUNCTIONS MANUEL MERIDA-ANGULO´ AND KOEN THAS Abstract. In [19] it was explained how one can naturally associate a Deit- mar scheme (which is a scheme defined over the field with one element, F1) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over F1 such as com- binatorial F1-projective and F1-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit., and show that Deitmar schemes which are defined by finite trees (with possible end points) are “defined over F1” in Kurokawa’s sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all as- sociated Fq-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spec- trum of degrees, and which is thus very different than Ihara’s zeta function (which is trivial in this case). Using a process called “surgery,” we show that one can determine the zeta function of a general loose graph and its associated { Deitmar, Grothendieck }-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of exam- ples of loose graphs, and introduce the Grothendieck ring of F1-schemes along the way in order to perform the calculations.
    [Show full text]
  • The Physics and Computational Exploration of Zeta and L-Functions Chris King Feb 2016 Genotype 1.1.5
    The Physics and Computational Exploration of Zeta and L-functions Chris King Feb 2016 Genotype 1.1.5 Abstract: This article presents a spectrum of 4-D global portraits of a diversity of zeta and L- functions, using currently devised numerical methods and explores the implications of these functions in enriching the understanding of diverse areas in physics, from thermodynamics, and phase transitions, through quantum chaos to cosmology. The Riemann hypothesis is explored from both sides of the divide, comparing cases where the hypothesis remains unproven, such as the Riemann zeta function, with cases where it has been proven true, such as Selberg zeta functions. The Conspiracy of Classical Zeta and L-functions The Riemann zeta function provides the archetype for a vast array of complex functions whose fundamental basis is a deep relationship between products and sums, typified in the Riemann zeta ∞ −1 function by the Euler product formula ζ(z) = ∑n−z = ∏ (1− p−z ) Re(z)>1, in which a sum of n=1 p prime complex exponents of integers is equated with a product over the natural primes. Fig 1: A summary of Riemann's key results. Top left: zeta ζ and xi χ -30 ≤ x ≤ 12 |y| ≤ 27. Colour scheme on [0,1]: red logarithmic amplitude using log(1+|z|)/(1+log(1+|z|)), yellow/green (1-sin(angle(z)))/2, blue (1-cos(π(|z|))/2 with first peak at 1. (a) A functional equation giving an analytic continuation via the xi function, which shares the same non-trivial zeros. (b) A distribution for the non-trivial zeros, noting they probably all lie on x = 1/2.
    [Show full text]