Computing Stieltjes constants using complex integration

TL;DR

This paper introduces an efficient, high-precision algorithm for computing Stieltjes constants using complex integral representations, with rigorous error bounds and low complexity, implemented in the Arb library.

Contribution

It presents the first low-complexity, arbitrary-precision algorithm for Stieltjes constants based on integral representations and numerical contour evaluation.

Findings

Able to compute n(1) to 1000 digits in a minute for n up to 10^{100}

Provides new integral representations for n(v), (s), (s,v), and related functions

Achieves rigorous error bounds with efficient numerical methods

Abstract

The generalized Stieltjes constants $\gamma\_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ about its unique pole $s = 1$. In this work, we devise an efficient algorithm to compute these constants to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order~$n$. Our computations are based on an integral representation with a hyperbolic kernel that decays exponentially fast. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library. We can, for example, compute $\gamma\_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma\_n(v)$, $\zeta(s)$, $\zeta(s,v)$, some polygamma functions and the Lerch transcendent.