On the Erd\"{o}s flat polynomials problem, Chowla conjecture and Riemann Hypothesis

TL;DR

This paper proves that certain flat polynomial sequences cannot exist, confirming Erdős's conjectures, and shows that the Chowla conjecture implies the Riemann Hypothesis, linking number theory and dynamical systems.

Contribution

It establishes the non-existence of specific flat polynomials, confirms conjectures on Littlewood and Barker sequences, and connects the Chowla conjecture to the Riemann Hypothesis.

Findings

No square L^2-flat sequences of specified polynomials exist.

Erdős's conjectures on Littlewood polynomials are confirmed.

Chowla conjecture implies the Riemann Hypothesis.

Abstract

There are no square $L^2$-flat sequences of polynomials of the type $$\frac{1}{\sqrt q}( \epsilon_0 + \epsilon_1z + \epsilon_2z^2 + \cdots + \epsilon_{q-2}z^{q-2} +\epsilon_q z^{q-1}),$$ where for each $j,~~ 0 \leq j\leq q-1,~\epsilon_j = \pm 1$. It follows that Erd\"{o}s's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only finitely many Barker sequences. We further get that the spectrum of dynamical systems arising from continuous Morse sequences is singular. This settles an old question due to M. Keane. Applying our reasoning to the Liouville function we obtain that the popular Chowla conjecture on the %Bernouillicity normality of the Liouville function implies Riemann hypothesis.