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

TL;DR

This paper refines bounds on the Riemann zeta function's magnitude assuming the Riemann Hypothesis by using Fourier analysis and minorant functions, improving the constant C in Littlewood's classical estimate.

Contribution

It introduces a new approach using Fourier minorants to optimize the constant in Littlewood's bound on |(1/2+it)| under the Riemann Hypothesis.

Findings

Established that any C > ( 2)/2 is permissible in Littlewood's bound.

Connected the problem to Fourier minorants supported in a given interval.

Applied recent work of Carneiro and Vaaler to find the optimal minorant.

Abstract

In 1924 Littlewood showed that, assuming the Riemann Hypothesis, for large t there is a constant C such that |\zeta(1/2+it)| \ll \exp(C\log t/\log \log t). In this note we show how the problem of bounding |\zeta(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.