On the supremum of random Dirichlet polynomials
Volume 182 / 2007
Abstract
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.
