On the Erdös flat polynomials problem, Chowla conjecture and Riemann Hypothesis

Abstract

There are no square $L^2$-flat sequences of polynomials of the type $$\frac{1}{\sqrt{q}}({\epsilon }_0+{\epsilon }_{1}z+{\epsilon }_2{z}^2+\cdots +{\epsilon }_{q-2}{z}^{q-2}+{\epsilon }_q{z}^{q-1}),$$ where for each $j,0\le j\le q-1,{\epsilon }_j=\pm 1$. It follows that Erdös'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.

This is an extended version of my previous paper arXiv:1609.03435 with three new results and two more topics treated. We give a dynamical proof of the main result in arXiv:1609.03435 which assert that Erdös conjectures holds. As a consequence, we obtain that Chowla conjecture implies Riemann Hypothesis

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