I want to ask on the estimates of the sum $$ \sum_{n=1}^{\infty} \mu(n)M\Big(\frac{x}{n}\Big)=\frac{1}{2\pi i }\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{x^s}{s\zeta(s)^2}ds.$$ But it is little known to the sign-changes of the Mertens function. Additionally, 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 with the integrand $\zeta(s)^{-2}$.