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 $n \le 10^{100}$.
The Stieltjes Constants
The Stieltjes Constants are the coefficients in the Laurent expansion of the Riemann Zeta function g(z) about its simple pole at z= l. They can be represented as the limit of the difference between the sum of the first n terms of a series and the integral of its n-th term. The first 20 coefficients have been computed to 15 D using the Euler-Maclaurin method. As a by-product the sums of the series ~ T"=+ (-1)k(logk)"lk have been obtained to 15 D for n = 1 (l) 20.
Euler's Constant to 1271 Places
The value of Euler's or Mascheroni's constant Pi = lim"'. (1 + 2 + + (1/n) inn) has now been determined to 1271 decimal places, thus extending the previously known value of 328 places. A calculation of partial quotients and best rational approximations to Py was also made. 1. Historical Background. Euler's constant was, naturally enough, first evaluated by Leonhard Euler, and he obtained the value 0.577218 in 1735 [1]. By 1781 he had calculated it more accurately as 0.5772156649015325 [2]. The calculations were carried out more precisely by several later mathematicians, among them Gauss, who obtained Py = 0.57721566490153286060653. Various British mathematicians continued the effort [3], [4]; an excellent account of the work done on evaluation of Py before 1870 is given by Glaisher [5]. Finally, the famous mathematician-astronomer J. C. Adams [6] laboriously determined -y to 263 places. Adams thereby extended thes'vork of Shanks, who had obtained 110 places (101 of which were correct). Adams' result stood until 1952, when Wrench [9] calculated 328 decimal places. Although much work has been done trying to decide whether P is rational, the evaluation has not been carried out any more precisely. With the use of high-speed computers, the constants ir and e have been evaluated to many thousands of decimal places [11], [12]. A complete bibliography for ir appears in [11]. The evaluation of Py to many places is considerably more difficult. 2. Evaluation of y. The technique used here to calculate Py is essentially that used by Adams and earlier mathematicians. A complete derivation of the method is given by Knopp [7]. We use Euler's summation formula in the form