On the supremum of random Dirichlet polynomials

Mikhail Lifshits, Michel Weber

Research output

1 Citation (Scopus)

Abstract

We study the supremum of some random Dirichlet polynomials D N(t) = Σ N n=2=2 ε nd nn- σ-it, where (ε n) is a sequence of independent Rademacher random variables, the weights (d n) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials Σ n∈ετ = {2 ≤ n ≤ N:P +(n) ≤ p τ}, P +(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, double-script E sign sup t∈ℝN n=2 εn n-σ-it| ≈ N 1-σ/ logN The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.

Original languageEnglish
Pages (from-to)41-65
Number of pages25
JournalStudia Mathematica
Volume182
Issue number1
DOIs
Publication statusPublished - 7 Dec 2007

Scopus subject areas

  • Mathematics(all)

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