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Fast computation of the Stieltjes constants


Authors: José A. Adell and Alberto Lekuona
Journal: Math. Comp. 86 (2017), 2479-2492
MSC (2010): Primary 11M06; Secondary 60G51
DOI: https://doi.org/10.1090/mcom/3176
Published electronically: March 3, 2017
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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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Additional Information

José A. Adell
Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: adell@unizar.es

Alberto Lekuona
Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: lekuona@unizar.es

DOI: https://doi.org/10.1090/mcom/3176
Received by editor(s): April 21, 2015
Received by editor(s) in revised form: December 22, 2015, and April 26, 2016
Published electronically: March 3, 2017
Additional Notes: The authors were partially supported by Research Projects MTM2011-23998 and DGA, E-64, and by FEDER funds.
Article copyright: © Copyright 2017 American Mathematical Society