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  • [hal-01632280] On Veech’s proof of Sarnak’s theorem on the Möbius flow

    14 novembre 2017, par El Houcein El Abdalaoui
    We present Veech's proof of Sarnak's theorem on the Möbius flow which say that there is a unique admissible measure on the Möbius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the (...)
  • [hal-01634444] $R^2$ Dark Energy in the Laboratory

    14 novembre 2017, par Pierre Vanhove, Philippe Brax, Patrick Valageas
    We analyse the role, on large cosmological scales and laboratory experiments, of the leading curvature squared contributions to the low energy effective action of gravity. We argue for a natural relationship $c_0\lambda^2\simeq 1$ at low-energy between the $\cal R^2$ coefficients $c_0$ of the (...)
  • [hal-01625653] A conjecture which implies that there are infinitely many primes of the form n !+1

    13 novembre 2017, par Apoloniusz Tyszka
    Let f(6)=720, and let f(n+1)=f(n)! for every integer n \geq 6. For an integer n \geq 6, let \Lambda_n denote the following statement: if a system S \subseteq x_i!=x_j: 1 \leq i < j \leq n \cup x_i \cdot x_j=x_j+1: 1 \leq i < j \leq n-1 has at most finitely many solutions in integers (...)
  • [hal-01176039] Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals

    13 novembre 2017, par El Houcein El Abdalaoui, Mariusz Lemanczyk, Thierry De La Rue
    We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function (...)
  • [hal-01630363] Equations with powers of singular moduli

    13 novembre 2017, par Antonin Riffaut
    We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ), j(τ ′) such that the numbers 1, j(τ) m and j(τ ′) n are linearly dependent over Q for some positive integers m, (...)

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