Extremal bounds for Dirichlet polynomials with random multiplicative coefficients

Abstract

For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$D_N(t)=\frac{1}{\sqrt{N}}\sum _{n\le N}X(n)n^{it},$$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $\epsilon >0$ we show that, with high probability, $$\exp ((\log N{)}^{1/2-\epsilon })\ll \underset{|t|\le N^C}{\sup }|D_N(t)|\ll \exp ((\log N{)}^{1/2+\epsilon }).$$