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

2016/09/12 by El Houcein El Abdalaoui, Abdalaoui, el Houcein el
Mathematics · #05D99 (Primary) #37A05 #37A30 (Secondary) #42A05 #42A55 #Advanced Combinatorial Mathematics #Analytic Number Theory Research #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1609.03435

openalex publication_date 2016/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There are no square L²-flat sequences of polynomials of the type (1)/(√ q)( ε₀ + ε₁z + ε₂z² + ⋯ + ε_q-2z^q-2 +ε_q z^q-1), where for each j,~~ 0 ≤ j≤ q-1,~ε_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 %Bernouillicity normality of the Liouville function implies Riemann hypothesis.

Citations

• On a Conjecture of Littlewood and Idempotent Measures

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