On the supremum of random Dirichlet polynomials

Mikhail Lifshits; Michel Weber

Studia Mathematica, Tome 178 (2007), p. 41-65 / Harvested from The Polish Digital Mathematics Library

Résumé

We study the supremum of some random Dirichlet polynomials $D_{N} (t) = \sum_{n = 2}^{N} ε ₙ d ₙ n^{- σ - i t}$ , where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials $\sum_{n \in_{τ}} ε ₙ n^{- σ - i t}$ , $_{τ} = 2 \leq n \leq N : P ⁺ (n) \leq p_{τ}$ , 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, $s u p_{t \in ℝ} | \sum_{n = 2}^{N} ε ₙ n^{- σ - i t} | \approx (N^{1 - σ}) / (l o g N)$ . The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.

Publié le : 2007-01-01

Zbl 1124.30004

EUDML-ID : urn:eudml:doc:284720

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     title = {On the supremum of random Dirichlet polynomials},
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     pages = {41-65},
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Mikhail Lifshits; Michel Weber. On the supremum of random Dirichlet polynomials. Studia Mathematica, Tome 178 (2007) pp. 41-65. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm182-1-3/