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Algorithms for polylogarithms

The polylogarithm is defined for \(s, z \in \mathbb{C}\) with \(|z| < 1\) by

\[\operatorname{Li}_s(z) = \sum_{k=1}^{\infty} \frac{z^k}{k^s}\]

and for \(|z| \ge 1\) by analytic continuation, except for the singular point \(z = 1\).

Computation for small z

The power sum converges rapidly when \(|z| \ll 1\). To compute the series expansion with respect to \(s\), we substitute \(s \to s + x \in \mathbb{C}[[x]]\) and obtain

\[\operatorname{Li}_{s+x}(z) = \sum_{d=0}^{\infty} x^d \frac{(-1)^d}{d!} \sum_{k=1}^{\infty} T(k)\]

where

\[T(k) = \frac{z^k \log^d(k)}{k^s}.\]

The remainder term \(\left| \sum_{k=N}^{\infty} T(k) \right|\) is bounded via the following strategy, implemented in mag_polylog_tail().

Denote the terms by \(T(k)\). We pick a nonincreasing function \(U(k)\) such that

\[\frac{T(k+1)}{T(k)} = z \left(\frac{k}{k+1}\right)^s \left( \frac{\log(k+1)}{\log(k)} \right)^d \le U(k).\]

Then, as soon as \(U(N) < 1\),

\[\sum_{k=N}^{\infty} T(k) \le T(N) \sum_{k=0}^{\infty} U(N)^k = \frac{T(N)}{1 - U(N)}.\]

In particular, we take

\[U(k) = z \; B(k, \max(0, -s)) \; B(k \log(k), d)\]

where \(B(m,n) = (1 + 1/m)^n\). This follows from the bounds

\[\begin{split}\left(\frac{k}{k+1}\right)^{s} \le \begin{cases} 1 & \text{if } s \ge 0 \\ (1 + 1/k)^{-s} & \text{if } s < 0. \end{cases}\end{split}\]

and

\[\left( \frac{\log(k+1)}{\log(k)} \right)^d \le \left(1 + \frac{1}{k \log(k)}\right)^d.\]

Expansion for general z

For general complex \(s, z\), we write the polylogarithm as a sum of two Hurwitz zeta functions

\[\operatorname{Li}_s(z) = \frac{\Gamma(v)}{(2\pi)^v} \left[ i^v \zeta \left(v, \frac{1}{2} + \frac{\log(-z)}{2\pi i}\right) + i^{-v} \zeta \left(v, \frac{1}{2} - \frac{\log(-z)}{2\pi i}\right) \right]\]

in which \(s = 1-v\). With the principal branch of \(\log(-z)\), we obtain the conventional analytic continuation of the polylogarithm with a branch cut on \(z \in (1,+\infty)\).

To compute the series expansion with respect to \(v\), we substitute \(v \to v + x \in \mathbb{C}[[x]]\) in this formula (at the end of the computation, we map \(x \to -x\) to obtain the power series for \(\operatorname{Li}_{s+x}(z)\)). The right hand side becomes

\[\Gamma(v+x) [E_1 Z_1 + E_2 Z_2]\]

where \(E_1 = (i/(2 \pi))^{v+x}\), \(Z_1 = \zeta(v+x,\ldots)\), \(E_2 = (1/(2 \pi i))^{v+x}\), \(Z_2 = \zeta(v+x,\ldots)\).

When \(v = 1\), the \(Z_1\) and \(Z_2\) terms become Laurent series with a leading \(1/x\) term. In this case, we compute the deflated series \(\tilde Z_1, \tilde Z_2 = \zeta(x,\ldots) - 1/x\). Then

\[E_1 Z_1 + E_2 Z_2 = (E_1 + E_2)/x + E_1 \tilde Z_1 + E_2 \tilde Z_2.\]

Note that \((E_1 + E_2) / x\) is a power series, since the constant term in \(E_1 + E_2\) is zero when \(v = 1\). So we simply compute one extra derivative of both \(E_1\) and \(E_2\), and shift them one step. When \(v = 0, -1, -2, \ldots\), the \(\Gamma(v+x)\) prefactor has a pole. In this case, we proceed analogously and formally multiply \(x \, \Gamma(v+x)\) with \([E_1 Z_1 + E_2 Z_2] / x\).

Note that the formal cancellation only works when the order \(s\) (or \(v\)) is an exact integer: it is not currently possible to use this method when \(s\) is a small ball containing any of \(0, 1, 2, \ldots\) (then the result becomes indeterminate).

The Hurwitz zeta method becomes inefficient when \(|z| \to 0\) (it gives an indeterminate result when \(z = 0\)). This is not a problem since we just use the defining series for the polylogarithm in that region. It also becomes inefficient when \(|z| \to \infty\), for which an asymptotic expansion would better.