Extremal bounds for Dirichlet polynomials with random multiplicative coefficients
Volume 272 / 2023
Studia Mathematica 272 (2023), 59-80
MSC: Primary 11K65; Secondary 11N56.
DOI: 10.4064/sm220829-6-3
Published online: 27 April 2023
Abstract
For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = \frac1{\sqrt{N}} \sum_{n \leq N} X(n) n^{it},$$ with $t$ in various ranges. In particular, for fixed $C \gt 0$ and any small $\varepsilon \gt 0$ we show that, with high probability, $$\exp ( (\log N)^{1/2-\varepsilon})\ll \sup_{|t| \leq N^C} |D_N(t)| \ll \exp ( (\log N)^{1/2+\varepsilon }).$$
