Fast Method for Locating Peak Values of the Riemann Zeta Function on the Critical Line

Abstract

In this paper a new algorithm RS-PEAK will be presented for locating peak values of the Riemann zeta function on the critical line. The method based on earlier results of Andrew M. Odlyzko, Tadej Kotnik, and on a recently achieved results of solving simultaneous Diophantine approximation problems. Until 2014 only a few hundred values were known where the Riemann-Siegel Z-function (i.e: Z(t)) larger than 1000, mainly found by Ghaith Ayesh Hiary and Jonathan Bober. By applying the algorithm RS-PEAK thousands of large values can be produced where |Z(t)| > 1000 within a few hours on a single desktop PC. The aim of this paper is to describe the RS-PEAK algorithm by which many large values of Z(t) can be generated in order to be able to reveal new behaviours of the Riemann zeta function. Using RS-PEAK more than 20 000 values had been found during a two weeks period where |Z(t)| > 1000. The largest known Z(t) values are presented where log |Z(t)|/log(t) > 32=205.