Riemann’s zeta function and finite Dirichlet series
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Yu. V. Matiyasevich
Translated by: the author - St. Petersburg Math. J. 27 (2016), 985-1002
- DOI: https://doi.org/10.1090/spmj/1431
- Published electronically: September 30, 2016
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Abstract:
The paper describes computer experiments for calculating zeros and values of Riemann’s zeta function and of its first derivative inside the critical strip and to the left of it with the help of finite Dirichlet series the coefficients of which are defined via initial nontrivial zeros of the zeta function.References
- Gleb Beliakov and Yuri Matiyasevich, A parallel algorithm for calculation of determinants and minors using arbitrary precision arithmetic, BIT 56 (2016), no. 1, 33–50. MR 3486452, DOI 10.1007/s10543-015-0547-z
- —, Zeroes of Riemann’s zeta function on the critical line with $40000$ decimal digits accuracy, Research Data Australia, http://hdl.handle.net/10536/DRO/DU:30056270, 2013.
- Gleb Beliakov and Yuri Matiyasevich, Approximation of Riemann’s zeta function by finite Dirichlet series: a multiprecision numerical approach, Exp. Math. 24 (2015), no. 2, 150–161. MR 3350522, DOI 10.1080/10586458.2014.976801
- F. Johansson, Arb, http://fredrikj.net/arb/.
- Yu. Matiyasevich, Finite Dirichlet series with prescribed zeroes, http://logic.pdmi.ras.ru/~yumat/ personaljournal/finitedirichlet.
- —, New conjectures about zeroes of Riemann’s zeta function, Depart. Math. Univ. Leicester, Research Reports MA12-03; http://www2.le.ac.uk/departments/mathematics/research/ research-reports-2/reports_2012/ma12-03, http://logic.pdmi.ras.ru/~yumat/talks/ leicester _2012/MA12_03Matiyasevich.pdf, 2012.
- —, Calculation of Riemann’s zeta function via interpolating determinants, Max Planck Instit. Math., Bonn, Preprint 2013-18; http://www.mpim-bonn.mpg.de/preblob/5368, http://logic.pdmi.ras.ru/~yumat/talks/bonn_2013/5368.pdf, 2013.
- B. Riemann, Über die Anzhal der Primzahlen unter einer gegebenen Grösse, Monatsber. Berlin. Akad. (1859), 671–680.
Bibliographic Information
- Yu. V. Matiyasevich
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, Saint Petersburg 191023, Russia
- Email: yumat@pdmi.ras.ru
- Received by editor(s): June 1, 2015
- Published electronically: September 30, 2016
- Additional Notes: The research was partially supported by the Ministry of Education and Science of the Russian Federation (Grant 14.Z50.31.0030).
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 985-1002
- MSC (2010): Primary 11M26; Secondary 11M06, 11M35, 11M41, 15A15, 11Y35
- DOI: https://doi.org/10.1090/spmj/1431
- MathSciNet review: 3589227
Dedicated: To the 70th anniversary of Sergey Vladimirovich Vostokov