$M(n)$ is Merten's function which is the sum of the Mobius function, $\mu(n).$
$$M(n)=\sum \mu(n).$$
Define the following functions:
$$ \Phi(n)= \sum M(n) $$
$$ \Psi(n)= \sum \Phi(n) $$
$$ \Omega(n)=\sum \Psi(n). $$
What do the plots of $\Phi(n)$ and $\Psi(n)$ and $\Omega(n)$ look like? (unanswered).
There is a relation between the zeta function and the mobius function:
$$ \frac{1}{\zeta(s)}=\sum_{n=1}^\infty \frac{\mu(n)}{n^s}. $$
Are the following related to $\zeta?$ $$\sum \frac{M(n)}{n^s}$$
$$ \sum \frac{\Phi(n)}{n^s} $$
$$ \sum \frac{\Psi(n)}{n^s} $$
$$ \sum \frac{\Omega(n)}{n^s}. $$
For a general function $f(n),$
what does the LHS look like? $$ ?=\sum \frac{f(n)}{n^s} .$$
$$M(x) = \sum_{n \le x} \mu(n)$$
For $|x| < 1$, $(1+x)^{-s} = \sum_{k = 0}^\infty {-s\choose k} x^k$
$$\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \mu(n) n^{-s} = s \int_1^\infty M(x) x^{-s-1}dx$$
$$ = M(n) (n^{-s}-(n+1)^{-s}) =-\sum_{k=1}^\infty {-s \choose k} \sum_{n=1}^\infty M(n) n^{-s-k} $$
If $M(x) = O(x^{1/2+\epsilon})$ then $ M(n) (n^{-s}-(n+1)^{-s})=\sum_{n=1}^\infty \mu(n) n^{-s} $ converges (so is analytic) for $\Re(s) > 1/2$ and the RH is true.
The converse is a PNT-like Tauberian atheorem : if the RH is true then $M(x) = O(x^{1/2+\epsilon})$ and $\sum_{n=1}^\infty M(n) n^{-s-1}$ converges for $\Re(s) > 1/2$.
Note the Mellin transform of $f(x)=\sum_{n=1}^\infty \mu(n) e^{-nx}$ is $\frac{\Gamma(s)}{\zeta(s)}$ and $\sum_{n=1}^\infty M(n)e^{-nx}=\frac{f(x)}{1-e^{-x}} \approx \frac{f(x)}{x}$. There are explicit formulas for $f(x)$ and $\frac{f(x)}{1-e^{-x}}$ in term of the zeros.