Low moments of Dirichlet series
In order to bound the mean value of multiplicative functions precise bounds for integrals like \[ I(q)=\int_{-1}^1\left| \sum_p\frac{a_p\log p}{p^{\sigma+it}}\right| ^qdt,\quad\text{and}\quad J(q)=\int_{-1}^1\left| \sum_{n\geq1}\frac{b_n}{n^{\sigma+it}}\right| ^qdt \] are needed. It can easily be deduced that \[ I(q)\ll(\sigma-1)^{1-q},\quad J(q)\ll(\sigma-1)^{1-q} \] for any \(2\leq q\leq4\) in the case when \(| a_p| \leq1\) for all prime \(p\) and \(| b_n| \leq1\) for all positive integer \(n\). The goal of the authors is to extend these estimations for \(1\leq q\leq 2\). However, they point out that not exactly these relations extend to this wider range of parameters but the following is true: \[ I(q)\ll(\sigma-1)^{-q/2},\quad J(q)\ll(\sigma-1)^{-q/2} \] for any \(1\leq q\leq2\).
scientific article; zbMATH DE number 6384134
scientific article