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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.

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    $\begingroup$ At first glance this seems unlikely to me because it doesn't pass a 'smell test'. The main issue from my perspective is that the desnity of twin primes is expected to be less than the density of primes and in fact the sum over the set of twin primes $\sum_{p\in T}\frac1p$ is known to converge (Brun's theorem) but the infinite product $\prod_2^\infty\left(1-\frac2p\right)$ diverges at exactly the same rate as the analogous product $\prod_2^\infty\left(1-\frac1p\right)$ does. $\endgroup$
    – Steven Stadnicki
    Commented Nov 20, 2022 at 17:59
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    $\begingroup$ @StevenStadnicki This is wrong, $\prod_{p\leq x}(1-1/p)$ diverges like $1/\log x$ while $\prod_{p\leq x}(1-2/p)$ diverges like a constant times $1/(\log x)^2$. Up to a constant this is compatible with Hardy-Littlewood conjectures. $\endgroup$
    – Wojowu
    Commented Nov 20, 2022 at 18:08
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    $\begingroup$ Not sure what the question is. The twin prime asymptotic is a conjecture based on some strong random model for the primes, for example that for $n$ taken uniformly in $[1,N]$ and $p\le \sqrt{N}$ the random variables $X_p = n\bmod p$ are independent. Independent means that the probability that both $n,n+2$ are primes becomes $\prod_{p\le \sqrt{N}} Pr(n \ne 0\bmod p,n+2\ne 0\bmod p)$ (the $X_p$ are not independent, this is an approximate model, useful to capture the "randomness" of the primes) $\endgroup$
    – reuns
    Commented Nov 20, 2022 at 18:48
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    $\begingroup$ @Wojowu Glah, thank you. I was translating to the sum via logs and blanked on the fact that the multiplicative factor of 2 would actually translate back to an exponential factor on the asymptotics. $\endgroup$
    – Steven Stadnicki
    Commented Nov 20, 2022 at 19:23
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    $\begingroup$ Primes show up in the definition of $\phi$. Twin primes don't show up in the definition of $\phi_2$. $\endgroup$
    – Gerry Myerson
    Commented Nov 20, 2022 at 22:26

1 Answer 1

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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.

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  • $\begingroup$ Note that I excluded $p=2$ in my definition of $ϕ_2(n)$ in formula (7) in my question above because including it results in a zero product when $n$ is even as is the case for $S_2(n)$ when $n$ is even, so perhaps this accounts for the Handbook's different definitions for odd and even $n$. Also, note that in my related definition of $N_k$ in formula (6) in my question above the product starts with $p_2=3$ instead of $p_1=2$, $\endgroup$
    – Steven Clark
    Commented Nov 20, 2022 at 22:25
  • $\begingroup$ V. A. Golubev's paper is available on mathnet.ru: mathnet.ru/php/…. On the first page, equation (2) defines $\varphi_s(n)$ for positive integers $s$ and $n$: it is $n\prod_{p_1 \mid (n,s)} (1 - 1/p_1)\prod_{p_2 \mid n, p_2 \nmid s} (1-2/p_2)$, where the products run over primes, so $\varphi_2(n)= (1/2)n\prod_{p \mid n, p>2} (1-2/p)$ for even $n$ (that's his equation (3)) and $\varphi_2(n)= n\prod_{p \mid n} (1-2/p)$ for odd $n$. There is a table of twin prime counts $\pi_2(x)$ in a table on the bottom of the 2nd page. $\endgroup$
    – KConrad
    Commented Nov 20, 2022 at 22:25
  • $\begingroup$ Golubev's table on the bottom of the 2nd page has correct counts for $\pi_2(x)$ (twin primes up to $x$), but in the row for counts of $\pi_4(x)$ (prime pairs $(p,p+4)$ where $p \leq x$) there are errors: the values of $\pi_4(j \cdot 10^4)$ for $j = 5, 6, 7, 8, 9$ are all too low by $1$. $\endgroup$
    – KConrad
    Commented Nov 20, 2022 at 22:34
  • $\begingroup$ Section 3 of Golubev's paper focuses on twin primes and he heuristically derives the standard conjecture $\pi_2(x) \sim cx/(\log x)^2$, where $c = 2\prod_{p>2} (1 - 1/(p-1)^2)$. He notes that this asymptotic conjecture goes back to Hardy and Littlewood (1923) but says he was unaware of their work during his own, and he had found the formula for $c$ independently in 1950. Nowadays this and more general conjectures about prime values of polynomials are special cases of the Bateman-Horn conjecture (1962). $\endgroup$
    – KConrad
    Commented Nov 20, 2022 at 22:44
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    $\begingroup$ If you look in Golubev's article, after working out equations (4), (5), and (6) he replaces $\varphi(n)$ and $\varphi_2(n)$ for even $n$ with $\pi(x)$ and $\pi_2(x)$, and $\prod_{p|n, p>2} (1 - 1/(p-1)^2)$ with $\prod_{p>2} (1 - 1/(p-1)^2)$, to obtain heuristically the conjectural asymptotic estimate on $\pi_2(x)$ from the prime number theorem for $\pi(x)$. That answers the OP's question about how the growth of $\varphi_2(n)$ (for even $n$) can be related to the twin prime conjecture. $\endgroup$
    – KConrad
    Commented Nov 21, 2022 at 1:10

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