Section: New Results

Computing Stieltjes constants using complex integration

Participant : Fredrik Johansson.

In [32], F. Johansson and I. Blagouchine devise an efficient algorithm to compute the generalized Stieltjes constants ${\gamma }_n\left(a\right)$ to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order $n$. 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.