Can the twin-prime conjecture be related to the growth of $\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)$? This question is based on this Math StackExchange question by user buja and my related Math StackExchange question which was asked about a month ago and is still unanswered.

I believe
$$\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)\tag{1}$$
is related to the twin-primes analogous to the way Euler's totient function
$$\phi(n)=n \prod\limits_{p|n}\left(1-\frac{1}{p}\right)\tag{2}$$
is related to the primes.


Question: Noting that the Euler totient function $\phi(n)$ defined in formula (2) above is related to the prime number theorem for arithmetic progressions and the Riemann hypothesis, I'm wondering if the twin-prime conjecture can be related to the growth of $\phi_2(n)$ defined in formula (1) above.


The remainder of this question clarifies the analogy I referred to previous to my question above.

The number of elements left after removing all primes up to $p_k$ and their multiples from the set $\left\{1,2,3,...p_k\#\right\}$ where $p_k\#=\prod\limits_{i=1}^k p_i$ is the primorial is given by
$$E_k=p_k\#\ \prod\limits_{i=1}^k \left(1-\frac{1}{p_i}\right)=\prod\limits_{i=1}^k \left(p_i-1\right)\tag{3}$$
which can be written in terms of the more general Euler totient function
$$\phi(n)=n \prod\limits_{p|n}\left(1-\frac{1}{p}\right)\tag{4}$$
as
$$E_k=\phi\left(p_k\#\right).\tag{5}$$

I believe the number of elements left after removing all multiples of primes up to $p_k$ and their associates (defined as pairs of numbers of the form $6 n\pm 1$) from the set $\left\{1,2,3,...p_k\#\right\}$ is given by
$$N_k=p_k\#\ \prod\limits_{i=2}^k \left(1-\frac{2}{p_i}\right)=2 \prod\limits_{i=2}^k \left(p_i-2\right)\tag{6}$$
which can be written in terms of the more general function
$$\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)\tag{7}$$
as
$$N_k=\phi_2\left(p_k\#\right).\tag{8}$$

I believe $\phi_2(n)$ is related to $\phi(n)$ as follows
$$\phi_2(n)=\sum\limits_{d|n} (-1)^{d-1}\ \mu(rad(d))\ \phi\left(\frac{n}{d}\right)\tag{9}$$
where $\mu(n)$ is the Möbius function and $rad(n)$ is the radical of an integer.

Note that $\mu(rad(n))=(-1)^{\nu(n)}$ where $\nu(n)$ is the number of distinct primes dividing $n$ (see OEIS entry A001221), so formula (9) above can also be evaluated as follows.
$$\phi_2(n)=\sum\limits_{d|n} (-1)^{d-1}\ (-1)^{\nu(d)}\ \phi\left(\frac{n}{d}\right)\tag{10}$$

My related question on Math StackExchange provides additional information which perhaps provides a bit more insight.
 A: This approach to trying to understand twin primes dates back at least to the 1950s. Sandor and Crstici's "Handbook of Number Theory II" mentions on page 289 in this context papers by V. A Golubev on this function, with references. In particular, he defined $\phi_2(n)$ as your function when $n$. The Handbook gives a slightly different definition for when $n$ is even, but I think there's a typo in their definition, and I can't say for certain what the issue is. Unfortunately, it looks like Golubev's papers are in Russian, so tracking them down and seeing exactly what is in them may be difficult. But this does suggest that the idea has at least been tried.
Note also that there's a closely related related function $S_k(n)$, as the number of sets of consecutive positive integers which are less than or equal to $n$, and all relatively prime to $n$. This is the Schemmel totient function. For $k=2$, it is essentially your function with $$S_2(n) = n\prod_{p|n} \left(1-\frac{2}{p}\right).$$ The Schemmel totient is of interest for reasons which are not necessarily connected with anything involving twin primes.
