Arithmetical Aspects of Beurling's Real Variable Reformulation of the Riemann Hypothesis

TL;DR

This paper explores arithmetical reformulations of Beurling's approach to the Riemann Hypothesis, providing elementary proofs and insights into convergence issues of Beurling functions.

Contribution

It introduces two arithmetical versions of the Nyman-Beurling equivalence, proved using classical number-theoretic methods, enhancing understanding of Beurling functions' convergence behavior.

Findings

Proved two arithmetical versions of the Nyman-Beurling equivalence.

Provided elementary, number-theoretic proofs of these equivalences.

Gained insights into the convergence phenomena of Beurling functions.

Abstract

The paper presents two arithmetical versions of the Nyman-Beurling equivalence with the Riemann hypothesis, proved by classical, quasi elementary, number-theoretic methods, based on an integrated version of the classical combinatorial identity for Moebius numbers. These proofs also give insight into the troublesome phenomenon that many natural sequences of Beurling functions tending to the indicator function of (0,1) both pointwise and in L1 norm do not converge in L2 norm.