Exponential sums of the M\"{o}bius function and flat polynomials

TL;DR

This paper explores the connection between elementary problems on specific analytic polynomials and the Riemann hypothesis, linking flat polynomial problems to deep number theory conjectures.

Contribution

It identifies a particular case of Erdős's flat polynomial problem that implies the Riemann hypothesis and provides a partial converse under additional conditions.

Findings

Elementary polynomial problem implies Riemann hypothesis

Connection established between flat polynomials and number theory

Partial converse proven under extra conditions

Abstract

We present an elementary problem on analytic polynomials with coefficients $\pm 1$ or in $\{0,\pm 1 \}$ which implies Riemann hypothesis. It is turns out that this problem is a particular case of the weak form of a flat polynomials problem asked by Erd\"os in his 1957's paper. We provide also under an extra-condition the converse.