Computing Stieltjes constants using complex integration

The Stieltjes constants γn are the coefficients appearing in the Laurent series of the Riemann zeta function at s = 1. We give a simple and efficient method to compute a p-bit approximation of γn with rigorous error bounds. Starting from an integral representation due to Blagouchine, we shift the contour to eliminate cancellation. The integral is then evaluated numerically in ball arithmetic using the Petras algorithm, with the use of a Taylor expansion for bounds near the saddle point. This appears to be the first algorithm for Stieltjes constants with uniformly low complexity with respect to both n and p. An implementation is provided in the Arb library. We can, for example, compute γn to 1000 digits in a minute for any n ≤ 10.