The attached paper claims on top of page 4 that
\begin{align} Res(\zeta^2(s)\prod_{p|d}(1-p^{-s})^2\frac{x^{s-1}}{s}; s=1) = \frac{\varphi^2(d)}{d^2}(\log{x}+2\gamma-1)-\frac{2\varphi(d)}{d}\sum_{\delta|d}\frac{\mu(\delta)\log{\delta}}{\delta} \end{align}
Could someone help me with the proof? I tried the proof for few specific $d's$ and was able to see the relation, but could not see how $\varphi$ and sum over divisors come up for general $d$.
You meant $$\prod_{p|d}(1-p^{-s})^2$$ What is the Laurent expansion of $\zeta(s)^2$ at $s=1$ ? And that of $F(s)=\frac{x^{s-1}}{s}\prod_{p|d}(1-p^{-s})^2$ ? Thus what is the coef of $(s-1)^{-1}$ in the Laurent expansion of $\zeta(s)^2F(s)$ ?