Analytic Continuation of Harmonic Sums

TL;DR

Addresses the numerical evaluation of nested harmonic sums $S_{k_1,k_2, dots}(N)$ for complex $N$, motivated by Mellin-space methods in perturbative QCD and DGLAP/coefficient-function calculations. It develops an analytic continuation framework that expresses sums in terms of their large-$|N|$ behavior via Hurwitz zeta functions, and then constructs fast, recursive expansions from depth-1 sums to higher depths. Special treatment is given to sums with leading index 1, ensuring logarithmic growth can be handled within the same expansion framework. For cases where |N| is not large, the method shifts to larger arguments and recovers accurate results with controlled corrections. The authors also provide code implementations and explicit high-weight expansions to facilitate practical NNLO calculations.

Abstract

We present a method for calculating any (nested) harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) Hurwitz zeta functions, which allows a harmonic sum to be calculated as an expansion valid for large values of its argument. A program for implementing this method is also provided.