High precision computation of Riemann’s zeta function by the Riemann-Siegel formula, I
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- by J. Arias de Reyna;
- Math. Comp. 80 (2011), 995-1009
- DOI: https://doi.org/10.1090/S0025-5718-2010-02426-3
- Published electronically: September 24, 2010
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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.References
- J. Arias de Reyna, High precision computation of Riemann’s Zeta function by the Riemann-Siegel formula, II (to appear).
- M. V. Berry, The Riemann-Siegel expansion for the zeta function: high orders and remainders, Proc. Roy. Soc. London Ser. A 450 (1995), no. 1939, 439–462. MR 1349513, DOI 10.1098/rspa.1995.0093
- K. Chandrasekharan, Introduction to analytic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 148, Springer-Verlag New York, Inc., New York, 1968. MR 249348
- W. Gabcke, Neue Herleitung und explizite Restabschätzung der Riemann-Siegel Formel. Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Georg-August-Universität zu Göttingen, Göttingen, 1979.
- W. F. Galway, Analytic computation of the prime counting function, Dissertation, Urbana, Illinois. http://www.math.uiuc.edu/~galway/PhD_Thesis/
- X. Gourdon, The $10^{13}$ first zeros of the Riemann Zeta function, and zeros computation at very large height. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf
- B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Akad. Berlin, 1859, 671–680. (Also in Riemann’s Gesammelte Werke \cite{Nah}.)
- Bernhard Riemann, Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge, Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], Suppl. 1, BSB B. G. Teubner Verlagsgesellschaft, Leipzig; Springer-Verlag, Berlin, 1990 (German). Based on the edition by Heinrich Weber and Richard Dedekind; Edited and with a preface by Raghavan Narasimhan. MR 1066697, DOI 10.1007/978-3-663-10149-9
- C. L. Siegel, Uber Riemann’s Nachlaß zur analytischen Zahlentheorie, Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik 2 (1932), 45–80. Reprinted in \cite{Siegel-Werke}, 1, 275–310.
- Carl Ludwig Siegel, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1966 (German). Herausgegeben von K. Chandrasekharan und H. Maass. MR 197270
- E. C. Titchmarsh, The zeros of the Riemann zeta-function, Proc. Roy. Soc. London, 151 (1935), 234–255, and 157 (1936), 261–263.
Bibliographic 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.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2772105