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Mathematics of Computation

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Computational estimation of the order of $\zeta (\frac{1}{2}+it)$

Author: Tadej Kotnik
Journal: Math. Comp. 73 (2004), 949-956
MSC (2000): Primary 11M06, 11Y60; Secondary 11Y35, 65A05
Published electronically: July 14, 2003
MathSciNet review: 2031417
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Abstract: The paper describes a search for increasingly large extrema (ILE) of $\left\vert \zeta (\frac{1}{2}+it)\right\vert $ in the range $0\leq t\leq 10^{13}$. For $ t\leq 10^{6}$, the complete set of ILE (57 of them) was determined. In total, 162 ILE were found, and they suggest that $\zeta (\frac{1}{2} +it)=\Omega (t^{2/\sqrt{\log t\,\log \log t}})$. There are several regular patterns in the location of ILE, and arguments for these regularities are presented. The paper concludes with a discussion of prospects for further computational progress.

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Additional Information

Tadej Kotnik
Affiliation: Faculty of Electrical Engineering, University of Ljubljana, SI-1000 Ljubljana, Slovenia

Keywords: Riemann's zeta function, critical line, Lindel\"{o}f's hypothesis
Received by editor(s): April 24, 2002
Received by editor(s) in revised form: October 21, 2002
Published electronically: July 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society