# Computing Stieltjes constants using complex integration

1 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : The generalized Stieltjes constants $\gamma_n(v)$ are, up to a simple scaling factor, the Laurent series coefficients of the Hurwitz zeta function $\zeta(s,v)$ 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 $\gamma_n(1)$ to 1000 digits in a minute for any $n$ up to $n=10^{100}$. We also provide other interesting integral representations for $\gamma_n(v)$, $\zeta(s)$, $\zeta(s,v)$, some polygamma functions and the Lerch transcendent.
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Journal articles
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Cited literature [32 references]

https://hal.inria.fr/hal-01758620
Contributor : Fredrik Johansson <>
Submitted on : Saturday, August 11, 2018 - 1:51:20 PM
Last modification on : Friday, December 20, 2019 - 2:21:59 PM

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stieltjes.pdf
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### Identifiers

• HAL Id : hal-01758620, version 3
• ARXIV : 1804.01679

### Citation

Fredrik Johansson, Iaroslav Blagouchine. Computing Stieltjes constants using complex integration. Mathematics of Computation, American Mathematical Society, 2019, 88 (318). ⟨hal-01758620v3⟩

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