On the supremum of random Dirichlet polynomials

Mikhail Lifshits, and Michel Weber. "On the supremum of random Dirichlet polynomials." Studia Mathematica 182.1 (2007): 41-65. <http://eudml.org/doc/284720>.

@article{MikhailLifshits2007,
abstract = {We study the supremum of some random Dirichlet polynomials $D_\{N\}(t) = ∑_\{n=2\}^\{N\} εₙdₙn^\{-σ-it\}$, 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 $∑_\{n∈ _\{τ\}\} εₙn^\{-σ-it\}$, $_\{τ\} = \{2 ≤ n ≤ N : P⁺(n) ≤ p_\{τ\}\}$, 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, $ sup_\{t∈ ℝ\} |∑_\{n=2\}^\{N\} εₙn^\{-σ-it\}| ≈ (N^\{1-σ\})/(log N)$. The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.},
author = {Mikhail Lifshits, Michel Weber},
journal = {Studia Mathematica},
keywords = {Dirichlet polynomials; Rademacher random variables; metric entropy method},
language = {eng},
number = {1},
pages = {41-65},
title = {On the supremum of random Dirichlet polynomials},
url = {http://eudml.org/doc/284720},
volume = {182},
year = {2007},
}

TY - JOUR
AU - Mikhail Lifshits
AU - Michel Weber
TI - On the supremum of random Dirichlet polynomials
JO - Studia Mathematica
PY - 2007
VL - 182
IS - 1
SP - 41
EP - 65
AB - We study the supremum of some random Dirichlet polynomials $D_{N}(t) = ∑_{n=2}^{N} εₙdₙn^{-σ-it}$, 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 $∑_{n∈ _{τ}} εₙn^{-σ-it}$, $_{τ} = {2 ≤ n ≤ N : P⁺(n) ≤ p_{τ}}$, 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, $ sup_{t∈ ℝ} |∑_{n=2}^{N} εₙn^{-σ-it}| ≈ (N^{1-σ})/(log N)$. The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.
LA - eng
KW - Dirichlet polynomials; Rademacher random variables; metric entropy method
UR - http://eudml.org/doc/284720
ER -