On the supremum of random Dirichlet polynomials

Mikhail Lifshits¹ ; Michel Weber²

¹Department of Mathematics and Mechanics St. Petersburg State University Bibliotechnaya pl. 2 198504, Stary Peterhof, Russia
²Mathématique (IRMA) Université Louis-Pasteur et C.N.R.S. 7 rue René Descartes 67084 Strasbourg Cedex, France

Studia Mathematica, Tome 182 (2007) no. 1, pp. 41-65

Résumé

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.

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DOI : 10.4064/sm182-1-3

Keywords: study supremum random dirichlet polynomials sum varepsilon n sigma where varepsilon sequence independent rademacher random variables weights multiplicative sigma particular attention given polynomials sum cal tau varepsilon sigma cal tau tau being largest prime divisor obtain sharp upper lower bounds supremum expectation extend optimal estimate hal queff lec mathbb sup mathbb sum varepsilon sigma approx sigma log proofs entirely based methods stochastic processes particular metric entropy method