This implements the Polylogarithm and some related functions that were needed (Harmonic numbers, Stieltjes constants, and Bernoulli numbers and polynomials, and Euler numbers).

The code is aimed at calculating Li_s(z) for all (complex) s and z.

This is still a little experimental, but there is a fairly large test set that all works nicely.

Note that the aimed for accuracy is 1.0e-12 relative error, but that occasional errors as large as 1.0e-11 have been seen.

• polylog(s, z) the polylogarithm function

• bernoulli(n) Provides the first 59 Bernoulli numbers as exact rationals

• bernoulli(n,x) Provides the Bernoulli polynomials

• euler(n) Provides the first 61 Euler numbers as BigInt

• harmonic(n) Provides the Harmonic numbers

• harmonic(n,r) Provides the generalised Harmonic numbers

• stieltjes(n) Provides the first 10 Stieltjes constants (see Abramowitz and Stegun, 23.2.5), also known as the generalized Euler-Mascheroni constants.

• dirichlet_beta(z) Provides the Dirichlet beta function

julia> using Polylogarithms
julia> polylog(2.0, 1.0)
1.6449340668482273

Extended details of the algorithms being used here are provided at https://arxiv.org/abs/2010.09860.

Accuracy has been tested over a wide range of scenarios. It starts to falter for large $z$ and $\Re(s) > 8$.

According to naming conventions, this package should have been called Polylogarithm, but there is already an older package doing polylogarithms, https://github.com/Expander/polylogarithm but it's using C/CPP/Fortran bindings, and only appears to do s=2,3,4,5,6.

There is a new package PolyLog, which does some newer stuff for dilogarithms, so that might be for you if you care about that case.