Computing Extremely Large Values of the Riemann Zeta Function

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.

[1]  J RieleteH.J.,et al.  On the zeros of the Riemann zeta function in the critical strip III , 1983 .

[2]  Richard P. Brent,et al.  On the zeros of the Riemann zeta function in the critical strip , 1979 .

[3]  Ghaith Ayesh Hiary,et al.  Fast methods to compute the Riemann zeta function , 2007, 0711.5005.

[4]  Norbert Tihanyi,et al.  Distributed computing of simultaneous Diophantine approximation problems , 2014 .

[5]  Norbert Tihanyi,et al.  Efficient computing of n-dimensional simultaneous Diophantine approximation problems , 2013 .

[6]  David P. Anderson,et al.  BOINC: a system for public-resource computing and storage , 2004, Fifth IEEE/ACM International Workshop on Grid Computing.

[7]  Ghaith A. Hiary,et al.  New Computations of the Riemann Zeta Function on the Critical Line , 2016, Exp. Math..

[8]  R. Sherman Lehman,et al.  Separation of zeros of the Riemann zeta-function , 1966 .

[9]  Tadej Kotnik,et al.  Computational estimation of the order of zeta (1/2 + it) , 2003, Math. Comput..

[10]  Arnold Schönhage,et al.  Fast algorithms for multiple evaluations of the riemann zeta function , 1988 .

[11]  Péter Kacsuk,et al.  SZTAKI Desktop Grid (SZDG): A Flexible and Scalable Desktop Grid System , 2009, Journal of Grid Computing.

[12]  Xavier.  The 10 13 first zeros of the Riemann Zeta function , and zeros computation at very large height , 2004 .

[13]  Jean Bourgain,et al.  Decoupling, exponential sums and the Riemann zeta function , 2014, 1408.5794.

[14]  Norbert Tihanyi,et al.  Fast Method for Locating Peak Values of the Riemann Zeta Function on the Critical Line , 2014, 2014 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[15]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[16]  Ghaith Ayesh Hiary,et al.  A nearly-optimal method to compute the truncated theta function, its derivatives, and integrals , 2007, 0711.5002.