Exponential sums of the Möbius function and flat polynomials

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 Erdös in his 1957's paper. We provide also under an extra-condition the converse.

arXiv comment8 pages. The motivation of the study of the polynomials with Liouville or Möbius coefficients is added and the results is linked to Pólya's approach to the Riemann hypothesis and several results on the maximum of those polynomials under GRH due essentially to Hajela and Smith. Some misprints and notations are corrected