Computing Stieltjes constants using complex integration
This provides a practical tool for number theorists and mathematicians needing high-precision values of Stieltjes constants, solving a previously computationally expensive problem.
The authors present an efficient algorithm for computing generalized Stieltjes constants to arbitrary precision with rigorous error bounds, achieving low complexity with respect to the order n. For example, they can compute γ_n(1) to 1000 digits in a minute for any n up to 10^100.
The generalized Stieltjes constants $γ\_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $ζ(s,v)$ 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 $γ\_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $γ\_n(v)$, $ζ(s)$, $ζ(s,v)$, some polygamma functions and the Lerch transcendent.