This morning I was playing (using Wolfram Alpha online calculator) with series involving the Möbius function $\mu(n)$ and the so-called Fejér kernel, see if you need it this Wikipedia.
My conclusion is that my example, next Question, has mathematical meaning (notice that the form of my integrand is due that I believe that such example has mathematical meaning, that is defined on $[0,1]$).
I did some unfinished calculations and without justifications (from my calculations I think that some of the resulting identities hold, but also that maybe is very difficult to justify those).
Question. Is it possible to calculate a good approximation (justifying it), or can you express as a closed-form as a series involving particular values of special functions (justifying it) the integral $$\int_0^1\sum_{n=1}^\infty x^2\mu(n)\left(\frac{1-\cos(nx)}{n(1-\cos(x))}\right)dx\,?\tag{1}$$ Many thanks.
Thus I am asking about what manipulations from real analysis and convergence one can to perform to express $(1)$ with the sign of summation outside of the integral, and after simplify the result in terms of an approximation or as closed-forms of series of particular values of special functions (if it is feasible).