The paper summarizes the computation results pursuing peak values of the Riemann zeta function. The computing method is based on the RS-Peak algorithm by which we are able to solve simultaneous Diophantine approximation problems efficiently. The computation environment was served by the SZTAKI Desktop Grid operated by the Laboratory of Parallel and Distributed Systems at the Hungarian Academy of Sciences and the ATLAS supercomputing cluster of the Eötvös Loránd University, Budapest. We present the largest Riemann zeta value known till the end of 2016.
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Brent, R.P.: On the zeros of the Riemann Zeta function in the critical strip. Math. Comp. 33(148), 1361–1372 (1979)
Sherman Lehman, R.: Separation of zeros of the Riemann Zeta-Function. Math. Comp. 20, 523–541 (1966)
Hiary, G.A.: Fast methods to compute the Riemann Zeta function. Ann. Math. 174-2, 891–946 (2011)
Odlyzko, A.M., Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann Zeta function. Trans. Am. Math. Soc. 309, 797–809 (1988)
Gourdon, X.: The 1013-rst zeros of the Riemann Zeta function, and zeros computation at very large height. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf. Accessed 5 December 2016 (2004)
Tihanyi, N.: Fast method for locating peak values of the Riemann Zeta function on the critical line. In: Sixteenth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE Explorer. https://doi.org/10.1109/SYNASC.2014.20 (2014)
Odlyzko, A.M.: The 1020-th zero of the riemann zeta function and 175 million of its neighbors. http://www.dtc.umn.edu/∼odlyzko/unpublished/. Accessed 5 December 2016 (1992)
Bober, J.W., Hiary, G.A.: New computations of the Riemann Zeta function on the critical line. Exp. Math., 27, 1–13 (2016)
Kotnik, Tadej: Computational estimation of the order of ζ(1/2 + i t). Math. Comp. 73(246), 949–956 (2004)
Lenstra, A.K., Lenstra, H. Jr, Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261(4), 515–534 (1982)
Kovács, A., Tihanyi, N.: Efficient computing of n-dimensional simultaneous Diophantie approximation problems. Acta Univ. Sapientiae, Informatica 5-1, 16–34 (2013)
Tihanyi, N., Kovács, A., Szücs, Á.: Distributed computing of simultaneous Diophantine approximation problems. Stud. Univ. Babes-Bolyai Math. 59(4), 557–566 (2014)
Bourgain, J.: Decoupling, exponential sums and the Riemann Zeta function. J. Am. Math. Soc. 30 (1), 205–224 (2017)
Kacsuk, P., Kovács, J., Farkas, Z., et al.: SZTAKI desktop grid (SZDG): a flexible and scalable desktop grid system. Journal of Grid Computing, Special Issue: Volunteer Computing and Desktop Grids 7(4), 439–461 (2009)
Anderson, D.P.: BOINC: a system for public-resource computing and storage. In: Proceedings of the 5th IEEE/ACM International Workshop on Grid Computing (GRID ’04). IEEE Computer Society, Washington, DC, USA
The SZTAKI desktop grid BOINC project. http://szdg.lpds.sztaki.hu/szdg. Accessed 5 December 2016
Peak performance of intel CPU’s. http://download.intel.com/support/processors/xeon/sb/xeon_5500.pdf. Accessed 5 December 2016
van de Lune, J., te Riele, H.J.J., Winter, D.T.: On the zeros of the Riemann Zeta function in the critical strip. IV. Math. Comp. 46, 667–681 (1986)
Hiary, G.A.: A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals. Annals of Math. 174, 859–889 (2011)
The authors gratefully acknowledge the constructive comments of Ghaith A. Hiary. We would like to thank the valuable suggestions for the anonymous reviewers. The authors would like to thank the opportunity for accessing to the ATLAS Super Cluster operating at Eötvös Loránd University and for using the capacity of SZTAKI Desktop Grid project operated by the Laboratory of Parallel and Distributed Systems in the Institute for Computer Science and Control of the Hungarian Academy of Sciences.
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Tihanyi, N., Kovács, A. & Kovács, J. Computing Extremely Large Values of the Riemann Zeta Function. J Grid Computing 15, 527–534 (2017). https://doi.org/10.1007/s10723-017-9416-0
- Riemann zeta function
- Distributed computing
- Large Z(t) values