Computing Stieltjes constants using complex integration
The Stieltjes constants $\gamma\_n$ 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 $\gamma\_n$ 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 $\gamma\_n$ to 1000 digits in a minute for any $n \le 10^{100}$.
An integral representation of the generalized Euler-Mascheroni constants
The limit series for the Euler-Mascheroni constants is represented as an integral. Using this new representation, the first 200 values are computed along with assorted others up to 2000. The first 13 roots of gamma sub n, where n is a positive continuous variable, are also given.
The sum of like powers of the zeros of the Riemann zeta function
On analyse une methode de calcul de la somme σ r =Σp −r ou r est un entier superieur a 1, la somme etant prise sur tous les zeros complexes de la fonction zeta de Riemann