I would like to know the asymptotic behaviour at large $n$ for $t\in\mathbb{R}$, $t\neq0$ of the following function: \begin{align*} A_n(t)&=\sum_{q=\frac{a}{b}\in \mathbb{Q}^+|\gcd(a,b)=1 \& ab=n}q^{it} \\ &=2^{\omega(n)}\prod_{i=1}^{\omega(n)}\cos(t\log p_i^{v_i}) \end{align*} where we have used the prime number decomposition of $q=\prod_{i}^{\omega(n)}p_i^{v_i}$ and $n=\prod_{i}^{\omega(n)}p_i^{|v_i|}$, and $\omega(q)=\omega(n)$ is their number of distinct prime factors.

On what can I rely to answer such question ? I can only notice: $$|A_n(t)|\leq A_n(0)=2^{\omega(n)}\sim 2^{\log\log n}, \quad n\gg 1$$ The aim is to find information about the convergence of the associated Dirichlet series $\sum_{n\geq 1}\frac{A_n(t)}{n^s}$ where $s\in \mathbb{R}$. The above remark leads to the absolute convergence for $s>1$, can we say better?

EDIT: As pointed out below, I got this expression writing $\frac{|\zeta(s+it)|^2}{\zeta(2s)}$ as a Dirichlet series.