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High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I


Author: J. Arias de Reyna
Journal: Math. Comp. 80 (2011), 995-1009
MSC (2010): Primary 11M06, 11Y35; Secondary 65E05
DOI: https://doi.org/10.1090/S0025-5718-2010-02426-3
Published electronically: September 24, 2010
MathSciNet review: 2772105
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Abstract | References | Similar Articles | Additional Information

Abstract: We present rigorous and sharp bounds for the terms and remainder in the Riemann-Siegel formula (for a general argument, not necessarily on the critical line). This allows for the computation of $\zeta (s)$ and $Z(t)$ to high precision. We also derive the Riemann-Siegel formula in a new and more direct way.


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References
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Additional Information

J. Arias de Reyna
Affiliation: Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain
ORCID: 0000-0003-3348-4374
Email: arias@us.es

Received by editor(s): December 3, 2009
Received by editor(s) in revised form: February 25, 2010
Published electronically: September 24, 2010
Additional Notes: The author was supported by grant MTM2009-08934.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.