GRHpack Home Page


These are instructions for obtaining the software GRHpack.m, a small Mathematica package of functions which goes with "Equivalents of the Riemann hypothesis: Volume Two, Analytic Equivalents", by Kevin A. Broughan published by Cambridge University Press. There are more details concerning the package, its installation and functions in the "GRHpack mini-manual", which is included in the book as Appendix L, page 463.

To use the software having Mathematica version 10.0 or later is recommended. Since the software is in the form of a standard mathematica package, it should run under any operating system (Windows, MacOS, or Unix/Linux).

Requests for assistance and advice are welcome, as are bug reports and suggestions for new software. If you find GRHpack useful please let me know, and give references for any published work which references the software. The email address is given below.

The Functions of GRHpack:

CharacterConductor, CharacterOrder, CharacterTimes, CharacterValue, ChiNM, ChiNN, ComputeHeight, CountSmoothIntegers, CyclotomicProduct, DickmanRho, DirichletCharacterQ, DirichletCharacters, DirichletCoefficientProduct, DirichletCoefficients, DirichletInverse, DirichletProduct, DiscreteMeasure, EllipticCount, EvenCharacterQ, EvenFundamentalCharacter, ExplicitBombieriVinogradov, ExplicitGallagher, FormBound, FundamentalCharacterQ, FundamentalDiscriminant, FundamentalDiscriminants, GaussCharacterSum, InducedCharacter, InducedCharacters, InducedModulusQ, OddCharacterQ, OddFundamentalCharacter, PlotRealDirichletL, PrimitiveCharacterQ, PrimitiveCharacters, PrincipalCharacter, PrincipalCharacterQ, ProbPosDefQ, QuadraticCharacterQ, RealCharacterQ, RealCharacters, Riesz, RootNumber, SmallestEigenvalue, SmoothIntegerQ, TitchmarshRemainder, WeilStepFunction.

GRHpack errata:

Page references are to Appendix L of Equivalents of the Riemann Hypothesis, vol. Two: Analytic Equivalents, CUP (2017).
  • p463 line -2 and p464 line 1: replace each single backslash with a double backslash
  • p465 line 14: terms → term
  • p466 line 9: bf v → v, begin → being
  • p467 line -8: m → b
  • p468 line 14: product → LCM
  • p468 line -12: [y,f] → [y,f,{a,b}]
  • p468 line -13: → "y is a real number"
  • p468 line -11: → "f is a real function"
  • p468 insert before line -9: "{a,b} represents a real interval"
  • p469 line 16: char → D, P → char
  • p469 line 17: → "D>0 is a fundamental discriminant"
  • p469 line 18: → "char is a Dirichlet character"
  • p472 line -4: prq → pcq
  • p475 entry for Xi should be removed - it is part of RHpack

    To obtain a copy of GRHpack.m send an email request to with GRHpack.m in the title.

    Last modified: 29th March 2024.