I want to ask on the estimates of the sum $$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$ It seems that the estimate of the sum be $x\exp(-c\sqrt{\log x})$ for some positive constant $c$, but it is little known to the sign-changes of the Mertens function. Also, the sum can be represented as $$ \sum_{1 \leq n \leq x}\bigg( \sum_{n=ab}\mu(a)\mu(b) \bigg). $$ It is easy to find the condition that $(the \ inner \ sum)=0$ if every degree of the prime factors of $n$ is bigger than $1$, where $k(n)$ is the square-free kernel of an integer. But also the representation has some difficulties.

So I thought that it is more good to deal with the integral, then there have to be some informations on the integrations with the integrand $\zeta(s)^{-2}$. I want to ask the estimates of the sum or the informations on the integrals which involve the integrand $\zeta(s)^{-2}$.

Additionally, the exact expression of the complex integral above with the zeros of the Riemann zeta function is given by $$ 4+2\sum_{|\text{Re } \rho| <1}\frac{x^{\rho}}{\zeta'(\rho)\rho}\phi_{(0,\rho)}+\sum_{n=1}^{\infty}\text{Res}\bigg[\frac{x^s}{s\zeta(s)^2},-2n\bigg],$$ where $$\phi_{(0,\rho)}=\sum_{k=1}^{\infty}\bigg\{ \frac{\mu(k)}{k^{\rho}}-\frac{1}{\zeta'(\rho)}\log\Big(\frac{k+1}{k}\Big) \bigg\} .$$ (Ref: N. A. Carella, "Simple Zeros Of The Zeta Function", page 3, http://arxiv.org/ftp/arxiv/papers/1306/1306.0458.pdf)