ZetaPack: A system for the computation of zeta and L-functions with Mathematica
Kevin A. Broughan
Version: 6th July 2007
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 denitions 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 dierential 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
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 Denitions . . . 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 Denitions . . . 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 Denitions . . . 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 Denitions . . . 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 Denitions . . . 73
7.3 Functional equation and Euler product . . . 73
7.4 Computation of the rational function representation . . . 74
7.5 Verication 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 Denitions . . . 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 Denitions . . . 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
10 Artin L functions 89
10.1 Introduction . . . 89
10.2 Denitions . . . 89
10.3 Euler product and functional equation . . . 92
10.3.1 Finite primes . . . 93
10.3.2 Innite 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 Denitions . . . 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 Elliptic curve 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 Denitions . . . 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 Dedekind zeta function . . . 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 Denitions 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
Bibliography 145
Index 149
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Index
GL(n; R), 16 T-group, 104 AnalyticRank, 67 abelian extension, 90 abelian group character, 39 AbelianPolynomialSplitPrimes, 95 AlgebraicToPolynomial, 51 an irreducible representation, 90 ApplyBlowupSubstitution, 112 Arithmetic schemes, 144 Artin character, 91 Artin conductor, 92 Artin root number, 92 ArtinLSeries, 99 ArtinRootNumber, 99 Automorphic forms, 16 autonomous, 115 Backlund's method, 135 bad Gram point, 133 BadGramPointQ, 137 band, 123
band number, 123 basin of attraction, 116 basin of repulsion, 116 BlowupPadicIntegral, 111 Casimir operator, 17 center, 116
character conductor, 39
character of a representation, 90 character sum, 90
character table, 90 CharacterConductor, 44
CharacterTimes, 42 CharacterValue, 41 class number of a eld, 50 classes, 12
commensurator, 20 cone conditions, 105 ConeConditions, 110
CongruenceCosetRepresentatives, 81 CongruenceCusps, 81
CongruenceFundamentalDomain, 82 ConjugacyClasses, 96
critical line, 133 critical point, 115 critical strip, 133 CubicToWeierstrass, 65 CyclotomicExpansion, 36 CyclotomicProduct, 34 CyclotomicProductQ, 34 DataPolygon, 125 decomposition group, 91 DecompositionGroup, 97 DedekindProduct, 100 degree of an extension, 48 Dictionaries, 23
dimension of a group, 104 Dirichlet character, 39 DirichletCharacters, 41
DirichletCoecients, 44, 57, 61, 69, 80, 109 DirichletCoecients[Artin,..], 100
DirichletInverse, 32 DirichletProduct, 32 DirichletToZetas, 33 discriminant of a eld, 48 discriminant of a polynomial, 47 discriminant of an element, 47 149
discriminant of an ideal, 48 discriminant of n elements, 47
DisplayFunctionalEquation, 41, 57, 60, 70, 83 dynamical zeta functions, 144
EichlerSelbergTrace, 82 EisensteinG, 77
EllipticDiscriminant, 64 EllipticMinus, 69 EllipticPeriods, 65 EllipticPlus, 68 EllipticTimes, 68
Encyclopedic properties, 12 equlibrium point, 115 EulerMcLaren, 37 EulerMclaren, 136 EulerPolynomialQ, 33 ExpandDataInterval, 124 FactorModularMatrix, 82 eld polynomial, 48 FieldPolynomial, 51
FindBlowupSubstitutions, 112 FindFirstGramBlock, 137
FindFundamentalDiscriminants, 43 FindZeros, 136
focus, 116
Fourier expansion of a Maass form, 18 FriendlyGhostQ, 36
FrobeneousReciprocity, 99 Frobenius automorphism, 91 FrobeniusAutomorphism, 98 fundamental discriminant, 50 fundamental units, 50
FundamentalDiscriminantQ, 42 FundamentalQuadraticUnit, 54 Galois extension, 89
Galois group of a polynomial, 90 Galois group of an extension, 90 GaloisGroup, 96
GAP, 144
GeneralWeierstrassToWeierstrass, 65 geometric zeta functions, 144 GetAlphaBeta, 34
GetIntegerPoints, 66 GetTorsionPoints, 66
GetValue, 23
GhostPolynomial, 35 global Artin conductor, 92 Godement-Jacquet L-function, 21 GonekTheta, 139
good Gram point, 133 GoodGramPointQ, 137 Gram block, 134 Gram point, 133 Gram's law, 134 GramPoints, 137
GramsLawExceptions, 138 group of units, 49
group representation, 90 GUE hypothesis, 135 Hecke L-functions, 144 Hecke operator, 20
HeckeMinimalPolynomial, 79 HeckeOperatorMatrix, 79 holomorphic ow, 115 ideal class group, 50 IdealClassGroup, 53 IdealClassNumber, 53 IharaRegular, 86 IharaZeta, 86
ImaginaryQuadraticClassNumber, 54 index of a generator, 49
index of a number eld, 49 induced character, 91 induced modulus, 39 InducedModulusQ, 43 inertia group, 91 InertiaGroup, 97 inertial degree, 49 InertialDegrees, 55
inessential discriminant divisor, 49 innite prime, 92
InitializeLcalcGlobals, 135 Installation, 9
IntegerPositiveDeniteQ, 60 Jacquet's Whittaker function, 19 LehmerZeros, 141
length of a Gram block, 134
INDEX 151 LenstraEigenvector, 80
LfunctionQ, 24 limit cycle, 116
local Artin conductor, 92 local zeta function, 105 LocalArtinConductors, 99 LocalFactorFinite, 98 LocalFactorInnite, 98 LRootNumber, 44 LValue, 136
Maass wave forms, 144 Magma, 144
MakeEntry, 23, 109, 110 MakeL, 24, 40, 57, 60, 64 Maple, 144
maximal domain of existence, 115 MeromorphicProductQ, 37 MillerModularBasis, 79
minimum index of a number eld, 49 minimum polynomial, 47
ModularDelta, 77
ModularFormsDimension, 78 ModularFormsDimensions, 81 MonogenicQ, 56
MonomialConditionsQ, 111 MonomialPadicIntegral, 112 motivic L-functions, 15 MultiplicativeQ, 31 NearestNeighbour, 140 negative separatrix, 116 NewtonPolygon, 35 nilpotent group, 103 node, 116
norm of a fractional ideal, 49 norm of an element, 48 norm of an ideal, 48 normal extension, 89
normalized zero distance, 135 NormalizedSpacings, 140 NormalizedZero, 140 NumberFieldDegree, 52 NumberFieldIndex, 53 orbit, 115
orbital neighborhood, 123
outside, 118
p-adic analytic group, 104 p-adic inertia group, 91 p-adic integral, 104 p-group, 103
p-Sylow subgroup, 103
pair correlation conjecture, 135 PairCorrelation, 139
Pari, 144 perfect eld, 89 periodic orbit, 116 phase diagram, 115 phase portrait, 115 PlotComplexFlow, 125 PlotEllipticCurve, 67 PlotIntegerPoints, 68 PlotTorsionPoints, 67 PolynomialDiscriminant, 51 PolynomialFactorPrimes, 95 positive separatrix, 116 PrimeDecomposition, 56 PrimeReport, 55
primitive character, 39 PrimitiveCharacterQ, 43 PrimitiveElement, 52 principal character, 39 PrincipalCharacterQ, 42 PrintEntry, 24
pro-p-completion, 104 pro-p-group, 104 PuiseuxExpansions, 36
QuadraticClassNumberOne, 55 quotient representation, 92 ramication index, 49 RamicationGroups, 98 RamiedPrimeData, 97 RamiedPrimes, 97 ramies, 49
rank of a pro-p-group, 104 re-entrant separatrix, 123 RealCharacterQ, 43 RealPositiveDeniteQ, 60 RealQuadraticClassNumber, 54 regular Gram block, 134
regulator, 50
RemoveRedundantConstraints, 112 residually nite group, 104
Riemann-Siegel #-function, 133 Riemann-Siegel Z-function, 133 RiemannSiegel, 37
RiemannSiegelNu, 38 RiemannSiegelNuSeries, 38 root number, 92
RootNumberConductor, 69 Rosser's rule, 134
RossersRuleException, 138 saddle point, 115
Sage, 144
Sato-Tate conjecture, 71 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 SubeldLattice, 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
TuringVerifyRHTuring, 141 UnitaryQ, 34
unstable zero, 116 ValidateL, 25, 40
virtually soluble group, 104 Web site for ZetaPack, 9 web sites, 144
WeierstrassToMinimal, 66 ZeroCountE, 139
ZeroCountN, 138 ZetaPack, 2
ZetasToDirichlet, 32 ZF1-ZF4, 12