FLINT3.7.0-dev
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acb_dirichlet_lerch_phi, acb_dirichlet_lerch_phi_direct, acb_dirichlet_lerch_phi_integral

Synopsis#include <flint/acb_dirichlet.h>
void acb_dirichlet_lerch_phi(acb_t res, const acb_t z, const acb_t s, const acb_t a, slong prec)source
void acb_dirichlet_lerch_phi_direct(acb_t res, const acb_t z, const acb_t s, const acb_t a, slong prec)source
void acb_dirichlet_lerch_phi_integral(acb_t res, const acb_t z, const acb_t s, const acb_t a, slong prec)source

Set res to the Lerch transcendent \Phi(z,s,a) = \sum_{k=0}^{\infty} z^k (k+a)^{-s}, analytically continued to |z| \ge 1. The direct version sums the truncated series; the integral version uses a Hankel contour integral; the default version checks special cases that reduce to the Hurwitz zeta function or polylogarithms and otherwise chooses between the two.

The integral representation is

\Phi(z,s,a) = -\frac{\Gamma(1-s)}{2 \pi i} \int_C \frac{(-t)^{s-1} e^{-a t}}{1 - z e^{-t}} dt,

with the path deformed as needed to avoid the poles and branch cuts of the integrand.

Cite this page

The FLINT team, “acb_dirichlet_lerch_phi, acb_dirichlet_lerch_phi_direct, acb_dirichlet_lerch_phi_integral (acb_dirichlet.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_dirichlet/acb_dirichlet_lerch_phi.html (generated 2026-10-20).