Laws of the iterated logarithm for random Dirichlet series with general weights

Abstract

For each $s>0$, we consider a random Dirichlet series $X(s)={\sum }_{k\ge 1}{k}^{-1/2-s}{a}_k{\eta }_k$, where ${\eta }_1$, ${\eta }_2,\dots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k{)}_{k\ge 1}$ is a deterministic sequence of real numbers satisfying ${\sum }_{k\ge 1}{k}^{-1-2s}{a}_k^2<\infty$ for each $s>0$ and ${\sum }_{k\ge 1}{k}^{-1}{a}_k^2=\infty$. We investigate the almost-sure fluctuations of $X(s)$ as $s\to 0+$. Under these minimal assumptions, we construct examples exhibiting several non-standard forms of the law of the iterated logarithm (LIL) along suitable sequences: the normalization and the upper and lower limit constants may differ from their classical counterparts. We also show that a regular growth condition of the form ${\sum }_{k\le n}{k}^{-1}{a}_k^2\sim c(\log n{)}^{\beta }$, where $c,\beta >0$, is not by itself sufficient to ensure a standard LIL. Finally, under an additional counting condition controlling the frequency of indices at which the weights $a_k$ are comparatively large, we prove that $(2\mathrm{Var}[X(s)]\log \log (\mathrm{Var}[X(s)]){)}^{-1/2}X(s)$ has the almost-sure cluster set $[-1,1]$ as $s\to 0+$. The latter result is applied to several coefficient sequences of number-theoretic origin.

submitted for publication; 36 pages