Bandlimited approximations and estimates for the Riemann zeta-function

TL;DR

This paper derives explicit bounds for the argument of the Riemann zeta-function and its derivatives in the critical strip, using Fourier analysis and approximation theory under the Riemann hypothesis.

Contribution

It introduces a novel Fourier optimization approach to bound the zeta-function's argument, sharpening existing estimates and extending them off the critical line.

Findings

Explicit bounds for the argument of the zeta-function under RH

Improved error terms in zeta-function estimates

Development of Fourier optimization techniques for bandlimited approximations

Abstract

In this paper, we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even functions by bandlimited functions, optimizing the $L^1(\mathbb{R})-$error. Deriving explicit formulae for the Fourier transforms of such optimal approximations plays a crucial role in our approach.