Abstract : There are no square L 2-flat sequences of polynomials of the type 1 √ q (ǫ 0 + ǫ 1 z + ǫ 2 z 2 + · · · + ǫ q−2 z q−2 + ǫqz q−1), where for each j, 0 ≤ j ≤ q −1, ǫ j = ±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 normality of the Liouville function implies Riemann hypothesis.