Let $\{a(n)\}_{n\in\mathbb N}$ be a sequence of real number, suh that for any $C\in \mathbb{R}$ we have $$a(n)\ll_{C}n^{C}$$ My question : is how we can prove that the Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)^2}{n^s}$$ converges for all $s\in \mathbb{C}.$
Many thanks
Given an $s\in\mathbb{C}$, we have a $b$ so that $\left|a(n)\right|\le bn^{\mathrm{Re}(s/2)-1}$ (let $C=\mathrm{Re}(s/2)-1$). Then $$ \left|\frac{a(n)^2}{n^s}\right|\le\frac{b^2}{n^2} $$ Therefore, $$ \sum_{n=1}^\infty\frac{a(n)^2}{n^s} $$ converges absolutely by comparison.