The Mertens function $M(x)$ is defined as follows.
(1) $\quad M(x)=\sum\limits_{n=1}^x\mu(n)$
I've noticed the Merten's function can also be evaluated as follows which is related to OEIS entry A087003 and the Collatz conjecture.
(2) $\quad M(x)=\sum\limits_{n=\left\lfloor\frac{x-2}{4}\right\rfloor+1}^{\left\lfloor\frac{x-1}{2}\right\rfloor}\mu(2\,n+1)$
I've verified formula (2) above for the first $10,000$ positive integer values of $x$.
Question (1): Has formula (2) above been proven (or disproven) and if not, can it be?
I've read the following conjecture on the growth of $M(x)$ for any $\epsilon<1/2$ is equivalent to the Riemann hypothesis (see Weisstein, Eric W. "Mertens Conjecture." From MathWorld--A Wolfram Web Resource).
(3) $\quad M(x)=O\left(x^{1/2+\epsilon}\right)$
Question (2): Assuming formula (2) above is correct, does it show any promise with respect to improving upon the fastest known algorithm for computing $M(x)$ and/or the tightest known error bound on the growth of $M(x)$?
There is nothing special happening here. Let $$F(s) = \sum_{n=0}^\infty \mu(2n+1)(2n+1)^{-s}, \qquad \frac{1}{\zeta(s)} = (1-2^{-s}) F(s)$$ $$f(x) = \sum_{2n+1 \le x} \mu(2n+1), \qquad M(x) = f(x)-f(x/2)$$