This question assumes the following definitions.
(1) $\quad\pi(x)==\sum\limits_{p\le x}1\qquad\text{(prime counting function where $p\in P$ is a prime})$
(2) $\quad\pi_2(x)==\sum\limits_{p_2\le x}1\qquad\text{(twin-prime counting function where $p_2\in P_2$ is a twin-prime)}$
(3) $\quad\pi_{sg}(x)==\sum\limits_{p_{sg}\le x}1\qquad\text{(Sophie Germain prime counting function where }\\$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{$p_{sg}\in P_{sg}$ is a Sophie Germain prime)}$
(4) $\quad M(x)=\sum\limits_{n\le x}\mu[n]\qquad\text{(Mertens function)}$
I believe the prime counting functions defined in (1) to (3) above can be expressed in terms of Mertens function defined in (4) above as follows where
- $\nu(n)$ is the number of distinct primes $p$ dividing $n$,
- $\nu_2(n)$ is the number of distinct twin-primes $p_2$ dividing $n$, and
- $\nu_{sg}(n)$ is the number of distinct Sophie Germain primes $p_{sg}$ dividing $n$.
(5) $\quad\pi(x)==\sum\limits_{n\le x}\nu(n)\ M\left(\frac{x}{n}\right)$
(6) $\quad\pi_2(x)==\sum\limits_{n\le x}\nu_2(n)\ M\left(\frac{x}{n}\right)$
(7) $\quad\pi_{sg}(x)==\sum\limits_{n\le x}\nu_{sg}(n)\ M\left(\frac{x}{n}\right)$
Formulas (5) to (7) are related to the following more general conjectured relationship which I've tested on several additional functions over small ranges of $x$.
(8) $\quad f(x)=\sum\limits_{n\le x} a(n)=\sum\limits_{n\le x} b(n)\ M(\frac{x}{n})\,,\quad b(n)=\sum\limits_{k|n} a(n)$
Question: Can the conjectured relationship illustrated in (8) above be proven which would also imply the correctness of formulas (5) to (7) above?
