Explicit estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

TL;DR

This paper provides a precise estimate for sums involving the Möbius function related to the Nyman-Beurling criterion for the Riemann Hypothesis, using advanced tools from continued fractions and Fourier series.

Contribution

It introduces a sharp bound for sums related to the Riemann Hypothesis, with explicit exponents, improving upon previous trivial bounds.

Findings

The estimate is significantly sharper than trivial bounds.

The bound's exponent is explicitly determined.

Methods combine continued fractions and Fourier analysis.

Abstract

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis. These sums contain the M\"obius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the M\"obius function. The bound is smaller than the trivial bound - essentially the number of terms - by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.