Fast computation of the Stieltjes constants

Adell, José; Lekuona, Alberto

doi:10.1090/mcom/3176

Fast computation of the Stieltjes constants
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by José A. Adell and Alberto Lekuona PDF

Math. Comp. 86 (2017), 2479-2492 Request permission

Abstract:

We evaluate each Stieltjes constant $\gamma _m$ as a finite sum involving the first $m+1$ Bernoulli numbers $B_k$ and the first $m+1$ derivatives $(-1)^k\alpha _k$ of the alternating zeta function at the point 1. In turn, we compute each $\alpha _k$ in an efficient way by means of a series with geometric rate 1/3. The coefficients of such a series are bounded and slowly decrease to zero. The computational significance of the preceding results is also discussed.

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