TY - JOUR

T1 - On the supremum of random Dirichlet polynomials

AU - Lifshits, Mikhail

AU - Weber, Michel

PY - 2007/12/7

Y1 - 2007/12/7

N2 - 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∈ℝ|Σ N 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.

AB - 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∈ℝ|Σ N 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.

KW - Dirichlet polynomials

KW - Metric entropy method

KW - Rademacher random variables

UR - http://www.scopus.com/inward/record.url?scp=36649010806&partnerID=8YFLogxK

U2 - 10.4064/sm182-1-3

DO - 10.4064/sm182-1-3

M3 - Article

AN - SCOPUS:36649010806

VL - 182

SP - 41

EP - 65

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 1

ER -