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arXiv · 0908.2008

Bounding |ζ(1/2 + it)| on the Riemann hypothesis

Vorrapan Chandee·Kannan Soundararajan
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Abstract

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |ζ(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |ζ(1/2+it)| may be framed in terms of minorizing the function \log ((4+x^2)/x^2) by functions whose Fourier transforms are supported in a given interval, and drawing upon recent work of Carneiro and Vaaler we find the optimal such minorant. Thus we establish that any C> (\log 2)/2 is permissible in Littlewood's result.

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Vorrapan Chandee, Kannan Soundararajan. 2009-08-14. Bounding |ζ(1/2 + it)| on the Riemann hypothesis. https://doi.org/10.1112/blms%2Fbdq095

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