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The mantissa distribution of the primorial numbers

Bruno Massé;  Dominique Schneider — 2014

Acta Arithmetica

We show that the sequence of mantissas of the primorial numbers Pₙ, defined as the product of the first n prime numbers, is distributed following Benford's law. This is done by proving that the values of the first Chebyshev function at prime numbers are uniformly distributed modulo 1. We provide a convergence rate estimate. We also briefly treat some other sequences defined in the same way as Pₙ.

Ergodic averages with deterministic weights

Fabien Durand;  Dominique Schneider — 2002

Annales de l’institut Fourier

We study the convergence of the ergodic averages 1 N ∑ k = 0 N - 1 θ ( k ) f ∘ T u k where ( θ ( k ) ) k ∈ ℕ is a bounded sequence and ( u k ) k ∈ ℕ a strictly increasing sequence of integers such that Sup α ∈ ℝ | ∑ k = 0 N - 1 θ ( k ) exp ( 2 i π α u k ) | = O ( N δ ) for some δ < 1 . Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q -multiplicative sequence.

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