ZetaPack: A system for the computation of zeta and L-functions with Mathematica Kevin A. Broughan Version: 6th July 2007 2 Contents Contents 3 0.1 Preface . 9 0.2 Description . 9 0.3 Installation . 9 0.4 Acknowledgements . 10 0.5 Dedication . 10 1 Zeta and L-functions 11 1.1 Introduction . 11 1.2 Classes of zeta and L-functions . 12 1.2.1 The Selberg class . 12 1.2.2 Properties of the Selberg Class . 13 1.2.3 Selberg's conjectures . 14 1.2.4 Consequences of the Selberg de nitions and conjectures . 15 1.2.5 Functions in the Selberg class . 15 1.2.6 Dokchitser L-functions . 15 1.2.7 Functions in the Dokchitser class . 16 1.3 Automorphic forms for GL(n; R) ............................. 16 1.3.1 Iwasawa decomposition . 16 1.3.2 Algebras of di erential operators . 17 1.3.3 The power function . 17 1.3.4 Maass forms . 18 1.3.5 Fourier expansions . 18 1.3.6 Jacquet's Whittaker function . 19 1.3.7 Hecke operators . 20 1.3.8 Godement-Jacquet L-function . 21 1.3.9 Functional equation . 22 1.4 Dictionaries . 23 1.5 ZetaPack functions . 23 1.6 The L-function data type . 24 2 A database of elementary Dirichlet Series 27 2.1 Introduction . 27 2.2 Multiplicative Functions . 27 3 4 CONTENTS 2.3 Database summary . 29 2.4 Operations on Dirichlet series . 31 2.5 Inverting a Riemann zeta related Dirichlet series . 32 2.6 Euler Products . 33 2.7 Evaluating the Riemann zeta function . 37 2.8 Historical notes and further reading . 38 3 Dirichlet L functions 39 3.1 Introduction . 39 3.2 De nitions . 39 3.3 The functional equation and Euler product . 40 3.4 ZetaPack functions . 40 3.5 Conjectures . 45 3.5.1 The generalized Riemann hypothesis (GRH) . 45 3.5.2 No Siegel zero conjecture . 45 3.5.3 The ratios conjecture . 45 3.6 Historical notes and further reading . 45 4 Dedekind zeta functions 47 4.1 Introduction . 47 4.2 De nitions . 47 4.3 The functional equation and Euler Product . 50 4.4 Computation in algebraic number elds . 51 4.4.1 Arithmetic operations . 51 4.4.2 Mathematica functions for algebraic numbers: . 51 4.4.3 ZetaPack functions . 51 4.4.4 Number Field Parameters . 52 4.4.5 Class numbers . 52 4.4.6 Prime decomposition . 55 4.5 The Dedekind L function type . 56 4.6 Deriving the Dirichlet coecients . 57 4.7 Conjectures . 58 4.7.1 Class number one conjecture . 58 4.7.2 Regulator conjecture . 58 4.7.3 Siegel zero . 58 4.8 Historical notes and further reading . 58 5 Epstein zeta functions 59 5.1 Introduction . 59 5.2 De nitions . 59 5.3 Functional equation . 59 5.4 ZetaPack functions . 60 5.5 Arithmetic Applications . 61 5.6 Failure of the Riemann hypothesis for Epstein zeta functions . 62 5.7 Conjectures . 62 5.8 Historical notes and further reading . 62 CONTENTS 5 6 Hasse-Weil zeta functions 63 6.1 Introduction . 63 6.2 De nitions . 63 6.3 The functional equation and Euler Product . 63 6.4 Functions for elliptic curves . 64 6.4.1 Mathematica functions . 64 6.4.2 ZetaPack functions . 64 6.5 Conjectures . 70 6.5.1 The Birch and Swinnerton-Dyer conjectures . 70 6.5.2 The Riemann hypothesis . 70 6.5.3 The Sato-Tate conjecture . 71 6.6 Historical notes and further reading . 71 7 Zeta functions for ane and projective hypersurfaces 73 7.1 Introduction . 73 7.2 De nitions . 73 7.3 Functional equation and Euler product . 73 7.4 Computation of the rational function representation . 74 7.5 Veri cation of the Riemann hypothesis . 74 7.6 Conjectures . 74 7.7 Historical notes and further reading . 74 8 Modular forms for SL(2,Z) 75 8.1 Introduction . 75 8.2 De nitions . 75 8.2.1 Holomorphic modular forms . 75 8.2.2 Vector spaces of modular forms . 75 8.2.3 Non-holomorphic modular forms . 76 8.2.4 Hecke Theory . 76 8.3 The functional equation and Euler product . 76 8.4 Computing with modular forms . 76 8.4.1 Mathematica functions . ..