On the supremum of random Dirichlet polynomials

Volume 182 / 2007

Mikhail Lifshits, Michel Weber Studia Mathematica 182 (2007), 41-65 MSC: Primary 30B50, 26D05; Secondary 60G17. DOI: 10.4064/sm182-1-3


We study the supremum of some random Dirichlet polynomials $D_N(t)=\sum_{n=2}^N\varepsilon_n d_n n^{-\sigma - it}$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, the weights $(d_n)$ are multiplicative and $0\le \sigma <1/2$. Particular attention is given to the polynomials $\sum_{n\in {\cal E}_\tau}\varepsilon_n n^{-\sigma - it}$, ${\cal E}_\tau=\{2\le n\le N\!: \! P^+(n)\le p_\tau\}$, $P^+(n)$ 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, $$ {\mathbb E}\, \sup_{t \in \mathbb R} \Big|\sum_{n=2}^N \varepsilon_n n^{-\sigma - it}\Big| \approx {N^{1-\sigma }\over \log N}. $$ The proofs are entirely based on methods of stochastic processes, in particular the metric entropy method.


  • Mikhail LifshitsDepartment of Mathematics and Mechanics
    St. Petersburg State University
    Bibliotechnaya pl. 2
    198504, Stary Peterhof, Russia
  • Michel WeberMathématique (IRMA)
    Université Louis-Pasteur et C.N.R.S.
    7 rue René Descartes
    67084 Strasbourg Cedex, France

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