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An improved upper bound for the argument of the Riemann zeta-function on the critical line

Author: Timothy Trudgian
Journal: Math. Comp. 81 (2012), 1053-1061
MSC (2010): Primary 11M06; Secondary 11M26
Published electronically: August 25, 2011
MathSciNet review: 2869049
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Abstract: This paper concerns the function $ S(t)$, the argument of the Riemann zeta-function along the critical line. Improving on the method of Backlund, and taking into account the refinements of Rosser and McCurley it is proved that for sufficiently large $ t$,

$\displaystyle \vert S(t)\vert \leq 0.1013 \log t. $

Theorem 2 makes the above result explicit, viz. it enables one to select values of $ a$ and $ b$ such that, for $ t>t_{0}$,

$\displaystyle \vert S(t)\vert \leq a + b\log t. $

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Additional Information

Timothy Trudgian
Affiliation: Department of Mathematics and Computer Science, University of Lethbridge, Alberta, Canada, T1K 3M4

Keywords: Riemann zeta-function, convexity estimate
Received by editor(s): October 21, 2010
Received by editor(s) in revised form: February 23, 2011
Published electronically: August 25, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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