Let $G(x)=\frac{1}{2\pi\iota}\int\limits_{(3/4)}\frac{\Gamma(1+t)^2}{(4\pi^2x)^t}\frac{dt}{t}$. Here, integration is over line such that real part is $\frac{3}{4}$. We can prove below result by using Stirling Approximation. $G(x)<< \begin{cases} {1} & {x\leq 1}\\ {exp(-c\sqrt{x})}& {x>1}\end{cases}$ for some $c>0$.
My question is how to prove : $\sum\limits_{n\geq 1}\frac{1}{n}G(\frac{n^2}{x})=\frac{1}{2}log(x) +c_0 +\mathcal{O}(\frac{1}{x})$.
As I have tried, LHS = $\frac{1}{2\pi\iota}\int\limits_{(3/4)}(\frac{x}{4\pi^2})^t\Gamma(1+t)^2\zeta(2t+1)\frac{dt}{t}$.
What can I do further?