# Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis

## TL;DR

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

## Contribution

It introduces a novel, highly precise estimate for sums with the Möbius function in the context of the Nyman-Beurling criterion, advancing understanding of the Riemann Hypothesis.

## Key findings

- Estimate is remarkably sharp compared to previous bounds
- Utilizes innovative methods from continued fractions and Fourier series
- Enhances analytical tools for studying the Möbius function in number theory

## Abstract

We give an estimate for sums appearing in the Nyman-Beurling criterion for the Riemann Hypothesis containing the M\"obius function. The estimate is remarkably sharp in comparison to estimates of other sums containing the M\"obius function. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.