Supremum of random Dirichlet polynomials with sub-multiplicative coefficients

We study the supremum of random Dirichlet polynomials $D_N(s)= \sum_{n=1}^N \varepsilon_n d(n) n^{- s }$, where $(\varepsilon_n)$ is a sequence of independent Rademacher random variables, and $d$ is a sub-multiplicative function. The approach is gaussian and entirely based on comparison properties of Gaussian processes, with no use of the metric entropy method. As in preceding related works, the proof uses a sieve argument due to Queffélec.