Timeline for Can the twin-prime conjecture be related to the growth of $\phi_2(n)=n \prod\limits_{p>2\,\land\,p|n}\left(1-\frac{2}{p}\right)$?
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| Nov 21, 2022 at 1:10 | comment | added | KConrad | 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. | |
| Nov 20, 2022 at 22:44 | comment | added | KConrad | 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). | |
| Nov 20, 2022 at 22:34 | comment | added | KConrad | 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 $1$. | |
| Nov 20, 2022 at 22:25 | comment | added | KConrad | 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. | |
| Nov 20, 2022 at 22:25 | comment | added | Steven Clark | 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 $p_1=2$, | |
| Nov 20, 2022 at 20:53 | history | answered | JoshuaZ | CC BY-SA 4.0 |