On the erdös flat polynomials problem, chowla conjecture and riemann hypothesis

El Houcein El Abdalaoui (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem

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.

Mots clés: Erdös-Newman; ultraflat polynomials; and phrases Merit factor; flat polynomials; Möbius function; Chowla conjecture; Liouville function; Morse sequences; trum; Barker sequences; singular spec-; Banach-Rhoklin problem; Banach problem; simple Lebesgue spectrum; conjecture; Turyn-Golay's; digital transmission; Morse cocycle; Littlewood flatness problem; flatness problem

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]; Cryptographie et sécurité [cs.CR]; Théorie spectrale [math.SP]; Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

El Houcein El Abdalaoui. On the erdös flat polynomials problem, chowla conjecture and riemann hypothesis. 2017. ⟨hal-01430637v2⟩