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 ...... 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 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 zeta related Dirichlet series ...... 32 2.6 Euler Products ...... 33 2.7 Evaluating the ...... 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 ...... 40 3.4 ZetaPack functions ...... 40 3.5 Conjectures ...... 45 3.5.1 The generalized (GRH) ...... 45 3.5.2 No Siegel zero ...... 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 ...... 76 8.4.2 ZetaPack functions ...... 77 8.5 Conjectures ...... 83 8.5.1 Lenstra's conjecture ...... 83 8.5.2 The Riemann Hypothesis ...... 83 8.6 Historical notes and further reading ...... 83

9 Ihara-Selberg zeta functions 85 9.1 Introduction ...... 85 9.2 De nitions ...... 85 9.3 The functional equation and Euler product ...... 86 9.4 Functions for computing with graphs ...... 86 9.4.1 Mathematica functions: ...... 86 9.4.2 ZetaPack functions ...... 86 9.5 Conjectures for Ihara zeta functions ...... 87 9.6 Historical notes and further reading ...... 87 6 CONTENTS

10 Artin L functions 89 10.1 Introduction ...... 89 10.2 De nitions ...... 89 10.3 Euler product and functional equation ...... 92 10.3.1 Finite primes ...... 93 10.3.2 In nite primes ...... 93 10.3.3 Functional equation ...... 93 10.4 Properties of the Artin L-function ...... 94 10.5 Examples ...... 94 10.5.1 Quadratic and multi-quadratic eld examples ...... 94 10.5.2 Lenstra's abelian example ...... 94 10.5.3 Sneyder/Heillbron's example: x3 n ...... 95 10.5.4 Artin's example: the icosahedral eld ...... 95 10.6 ZetaPack functions ...... 95 10.7 Conjectures ...... 101 10.7.1 Artin's conjecture ...... 101 10.7.2 Dedekind's conjecture ...... 102 10.7.3 Langland's conjectures ...... 102 10.8 Historical notes and further reading ...... 102

11 Group zeta functions 103 11.1 Introduction ...... 103 11.2 De nitions ...... 103 11.3 The Dirichlet series and Euler products ...... 105 11.4 Examples ...... 106 11.4.1 Finite groups ...... 106 11.4.2 Finitely generated free groups ...... 106 11.4.3 Finitely generated free abelian group ...... 106 11.4.4 Discrete Heisenberg group ...... 106 11.4.5 Heisenberg Lie ring ...... 107 11.4.6 Free class-two nilpotent group on three generators ...... 107 11.4.7 Crystallographic (wallpaper) groups ...... 107 11.4.8 The group sl2(Z)...... 108 11.4.9 counter example ...... 108 11.5 ZetaPack functions ...... 109 11.6 Conjectures and problems ...... 113 11.6.1 Uniformity conjecture ...... 113 11.6.2 Functional equation problem ...... 113 11.6.3 Lie Ring problem ...... 113 11.7 Historical notes and further reading ...... 113

12 Phase portraits of L function ows 115 12.1 Introduction ...... 115 12.2 De nitions ...... 115 12.3 Topological properties ...... 117 12.4 Phase portraits for the zeta ow ...... 119 CONTENTS 7

12.5 Phase Portraits for the xi ow ...... 121 12.6 ZetaPack functions ...... 124 12.7 Examples ...... 126 12.7.1 Riemann zeta and xi functions ...... 126 12.7.2 Real primitive Dirichlet L function ...... 126 12.7.3 Complex primitive Dirichlet L function ...... 126 12.7.4 ...... 126 12.7.5 Ihara zeta function ...... 126 12.7.6 Artin L function ...... 126 12.7.7 Epstein zeta function ...... 126

13 Connections between classes 127 13.1 Basic zeta functions which have functional equations ...... 127 13.2 Abelian Dedekind zeta functions and Dirichlet L functions ...... 127 13.3 Dedekind zeta functions and Epstein zeta functions ...... 128 13.4 Dedekind zeta functions and Artin L functions ...... 129 13.5 Artin L functions and Epstein zeta functions ...... 129 13.6 Relative and non-relative Artin L-functions ...... 130 13.7 Modular forms and elliptic curves ...... 130 13.8 A Group zeta function and Dedekind zeta function ...... 131

14 Numerical Evaluation and Zero analysis 133 14.1 Introduction ...... 133 14.2 De nitions for the Riemann Zeta function ...... 133 14.3 Evaluation of L-functions ...... 135 14.4 Finding zeros on the critical line ...... 136 14.5 Gram points ...... 137 14.6 Zero enumeration in the critical strip ...... 138 14.7 Pair correlation and nearest neighbour statistics ...... 139 14.8 Lehmer's phenomenon ...... 141 14.9 Verifying the RH, GRH and GGRH ...... 141

15 Epilogue 143 15.1 Other zeta and L-functions ...... 144 15.1.1 Hecke L-functions ...... 144 15.1.2 Maass wave forms ...... 144 15.1.3 Zeta functions of arithmetic schemes ...... 144 15.1.4 Shantani zeta functions ...... 144 15.1.5 Zeta functions of geometric origin ...... 144 15.1.6 Dynamical zeta functions ...... 144 15.2 Other software and information sources for zeta and L functions ...... 144 15.2.1 GAP ...... 144 15.2.2 Magma ...... 144 15.2.3 Maple ...... 144 15.2.4 Pari ...... 144 15.2.5 Sage ...... 144 15.2.6 Web sites ...... 144 8 CONTENTS

Bibliography 145

Index 149 Bibliography

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GL(n; R), 16 CharacterTimes, 42 T-group, 104 CharacterValue, 41 class number of a eld, 50 AnalyticRank, 67 classes, 12 commensurator, 20 abelian extension, 90 cone conditions, 105 abelian group character, 39 ConeConditions, 110 AbelianPolynomialSplitPrimes, 95 CongruenceCosetRepresentatives, 81 AlgebraicToPolynomial, 51 CongruenceCusps, 81 an irreducible representation, 90 CongruenceFundamentalDomain, 82 ApplyBlowupSubstitution, 112 ConjugacyClasses, 96 Arithmetic schemes, 144 critical line, 133 Artin character, 91 critical point, 115 Artin conductor, 92 critical strip, 133 Artin root number, 92 CubicToWeierstrass, 65 ArtinLSeries, 99 CyclotomicExpansion, 36 ArtinRootNumber, 99 CyclotomicProduct, 34 Automorphic forms, 16 CyclotomicProductQ, 34 autonomous, 115 DataPolygon, 125 Backlund's method, 135 decomposition group, 91 bad Gram point, 133 DecompositionGroup, 97 BadGramPointQ, 137 DedekindProduct, 100 band, 123 degree of an extension, 48 band number, 123 Dictionaries, 23 basin of attraction, 116 dimension of a group, 104 basin of repulsion, 116 Dirichlet character, 39 BlowupPadicIntegral, 111 DirichletCharacters, 41 DirichletCoecients, 44, 57, 61, 69, 80, 109 Casimir operator, 17 DirichletCoecients[Artin,..], 100 center, 116 DirichletInverse, 32 character conductor, 39 DirichletProduct, 32 character of a representation, 90 DirichletToZetas, 33 , 90 discriminant of a eld, 48 character table, 90 discriminant of a , 47 CharacterConductor, 44 discriminant of an element, 47

149 150 INDEX discriminant of an ideal, 48 GetValue, 23 discriminant of n elements, 47 GhostPolynomial, 35 DisplayFunctionalEquation, 41, 57, 60, 70, 83 global Artin conductor, 92 dynamical zeta functions, 144 Godement-Jacquet L-function, 21 GonekTheta, 139 EichlerSelbergTrace, 82 good Gram point, 133 EisensteinG, 77 GoodGramPointQ, 137 EllipticDiscriminant, 64 Gram block, 134 EllipticMinus, 69 Gram point, 133 EllipticPeriods, 65 Gram's law, 134 EllipticPlus, 68 GramPoints, 137 EllipticTimes, 68 GramsLawExceptions, 138 Encyclopedic properties, 12 group of units, 49 equlibrium point, 115 group representation, 90 EulerMcLaren, 37 GUE hypothesis, 135 EulerMclaren, 136 EulerPolynomialQ, 33 Hecke L-functions, 144 ExpandDataInterval, 124 Hecke operator, 20 HeckeMinimalPolynomial, 79 FactorModularMatrix, 82 HeckeOperatorMatrix, 79 eld polynomial, 48 holomorphic ow, 115 FieldPolynomial, 51 FindBlowupSubstitutions, 112 , 50 FindFirstGramBlock, 137 IdealClassGroup, 53 FindFundamentalDiscriminants, 43 IdealClassNumber, 53 FindZeros, 136 IharaRegular, 86 focus, 116 IharaZeta, 86 Fourier expansion of a Maass form, 18 ImaginaryQuadraticClassNumber, 54 FriendlyGhostQ, 36 index of a generator, 49 FrobeneousReciprocity, 99 index of a number eld, 49 Frobenius automorphism, 91 induced character, 91 FrobeniusAutomorphism, 98 induced modulus, 39 , 50 InducedModulusQ, 43 fundamental units, 50 inertia group, 91 FundamentalDiscriminantQ, 42 InertiaGroup, 97 FundamentalQuadraticUnit, 54 inertial degree, 49 InertialDegrees, 55 Galois extension, 89 inessential discriminant divisor, 49 Galois group of a polynomial, 90 in nite prime, 92 Galois group of an extension, 90 InitializeLcalcGlobals, 135 GaloisGroup, 96 Installation, 9 GAP, 144 IntegerPositiveDe niteQ, 60 GeneralWeierstrassToWeierstrass, 65 geometric zeta functions, 144 Jacquet's Whittaker function, 19 GetAlphaBeta, 34 GetIntegerPoints, 66 LehmerZeros, 141 GetTorsionPoints, 66 length of a Gram block, 134 INDEX 151

LenstraEigenvector, 80 outside, 118 LfunctionQ, 24 limit cycle, 116 p-adic analytic group, 104 local Artin conductor, 92 p-adic inertia group, 91 , 105 p-adic integral, 104 LocalArtinConductors, 99 p-group, 103 LocalFactorFinite, 98 p-Sylow subgroup, 103 LocalFactorIn nite, 98 pair correlation conjecture, 135 LRootNumber, 44 PairCorrelation, 139 LValue, 136 Pari, 144 perfect eld, 89 Maass wave forms, 144 periodic orbit, 116 Magma, 144 phase diagram, 115 MakeEntry, 23, 109, 110 phase portrait, 115 MakeL, 24, 40, 57, 60, 64 PlotComplexFlow, 125 Maple, 144 PlotEllipticCurve, 67 maximal domain of existence, 115 PlotIntegerPoints, 68 MeromorphicProductQ, 37 PlotTorsionPoints, 67 MillerModularBasis, 79 PolynomialDiscriminant, 51 minimum index of a number eld, 49 PolynomialFactorPrimes, 95 minimum polynomial, 47 positive separatrix, 116 ModularDelta, 77 PrimeDecomposition, 56 ModularFormsDimension, 78 PrimeReport, 55 ModularFormsDimensions, 81 primitive character, 39 MonogenicQ, 56 PrimitiveCharacterQ, 43 MonomialConditionsQ, 111 PrimitiveElement, 52 MonomialPadicIntegral, 112 principal character, 39 motivic L-functions, 15 PrincipalCharacterQ, 42 MultiplicativeQ, 31 PrintEntry, 24 pro-p-completion, 104 NearestNeighbour, 140 pro-p-group, 104 negative separatrix, 116 PuiseuxExpansions, 36 NewtonPolygon, 35 nilpotent group, 103 QuadraticClassNumberOne, 55 node, 116 quotient representation, 92 norm of a fractional ideal, 49 norm of an element, 48 rami cation index, 49 norm of an ideal, 48 Rami cationGroups, 98 normal extension, 89 Rami edPrimeData, 97 normalized zero distance, 135 Rami edPrimes, 97 NormalizedSpacings, 140 rami es, 49 NormalizedZero, 140 rank of a pro-p-group, 104 NumberFieldDegree, 52 re-entrant separatrix, 123 NumberFieldIndex, 53 RealCharacterQ, 43 RealPositiveDe niteQ, 60 orbit, 115 RealQuadraticClassNumber, 54 orbital neighborhood, 123 regular Gram block, 134 152 INDEX regulator, 50 TuringVerifyRHTuring, 141 RemoveRedundantConstraints, 112 residually nite group, 104 UnitaryQ, 34 Riemann-Siegel #-function, 133 unstable zero, 116 Riemann-Siegel Z-function, 133 RiemannSiegel, 37 ValidateL, 25, 40 RiemannSiegelNu, 38 virtually soluble group, 104 RiemannSiegelNuSeries, 38 Web site for ZetaPack, 9 root number, 92 web sites, 144 RootNumberConductor, 69 WeierstrassToMinimal, 66 Rosser's rule, 134 RossersRuleException, 138 ZeroCountE, 139 ZeroCountN, 138 saddle point, 115 ZetaPack, 2 Sage, 144 ZetasToDirichlet, 32 Sato-Tate conjecture, 71 ZF1-ZF4, 12 SC1-SC6, 14 Selberg class, 12 Selberg degree, 13 Selberg properties, 13 Selberg's conjectures, 14 separable extension, 89 separable polynomial, 89 separatrix, 116 SetValue, 23 Shintani zeta functions, 144 signature, 49 SimplePadicIntegral, 111 singular point, 115 sink, 116 soluble group, 103 solvable group, 103 source, 116 splitting eld extension, 89 stable zero, 116 StandardModularBasis, 78 StandardSquare, 125 Sub eldLattice, 96 SubgroupLattice, 96 subrepresentation, 90 sum of representations, 90 trace of an element, 48 trajectory, 115 transit time, 116 Turing's method, 134 TuringVerifyRHBacklund, 141