I'm going to start by saying I'm mostly out of my depth here. I'm an amateur recreational mathematician. But I've been looking at the Twin Prime Conjecture lately, because it is so fascinating. Easy enough for an 8-year-old to understand, but perplexing mathematicians for millennia. As a science teacher, the whole preponderance-of-empirical-evidence thing seems reasonable enough... but of course, not for mathematicians.
I have no delusion I'll prove anything, but maybe I can contribute something useful? And I have a weird result that may or may not be useful, that I also can't explain.
In relation to the TPC, I've been working on a sieve that generates This OEIS sequence, the numbers for which $6k \pm 1$ are both prime. I'm trying to build off of the work of Dinculescu's various papers (one linked, others at the OEIS link). The TPC is the conjecture that this sequence is infinite.
What does this have to do with Euler products? I've been working with the following one:
$$\prod_{5 \leq p \leq n} \left(1-\frac2p\right)$$
Simple enough. It starts at 5 because it's looking at twin primes, and has a 2 in there because the sieve removes $\frac2p$ of $\mathbb{N}$ each pass.
There are a lot of constants that can be calculated using Euler products. The one I'm working with above isn't one of them, or even similar. So I poked around a bit. Merten's 3rd theorem says this:
$$\lim_{n \to \infty}\ln n\prod_{p \leq n} \left(1 - \frac1p \right) = e^{-\gamma}$$
This theorem seemed interesting to me because the Euler product portion is basically the leftovers in $\mathbb{N}$ after $n$ iterations of the Sieve of Eratosthenes. As $n \to \infty$, the "leftover" set is just $\mathbb{P}$. So, the Sieve of Eratosthenes diverges to zero (as we'd expect since the density of $\mathbb{P}$ is zero), but when multiplied by $\ln n$, it converges. That's a pretty cool trick!
For the heck of it, I tried multiplying the iterations of my product by $\ln n$ to see what came out, which was still divergence to zero, uninteresting. So I looked at Hardy and Littlewood and noticed their constant ends up with an $(\ln n)^2$ term when calculating $\pi_2(n)$. Since my product diverges to zero about twice as fast as Mertens's, I decided to try that out, and I got something that converged! And I tried to figure out if the value it converged to ($\approx 2.49726$) was a known constant. After some trial and error I found:
$$\lim_{n \to\infty} (\ln n)^2 \prod_{5 \leq p \leq n} \left(1-\frac2p\right) \approx 4\lambda$$
Where $\lambda$ is... the Golomb-Dickman constant, something I'd never heard of. (This is by empricial computer calculation only, but it converges to four digits with about 6 million primes.) Apparently $\lambda d$ is the (asymptotic) average number of digits in the largest prime factor a $d$-digit integer.
But! We come around full circle. The complement of that OEIS sequence is the subset of $\mathbb{N}$ for which $6k \pm 1$ are not twin primes. These are all the natural numbers of the form $6ab \pm a \pm b, 1 \leq a \leq b$. That fact was discovered by.... Solomon Golomb.
I have exactly zero clue as to whether what I've written here is particularly useful. I certainly can't explain it. But at least to me, it's pretty interesting. Has anyone here worked with this sequence, and the TPC, and might have an explanation for what seems like it's maybe just a cosmic coincidence? Is this a new result, or am I chasing rabbits others have already chased?
(Edited (1): Typo in the numerical result. After ~6M primes, my value has converged to $2.49726$, not $2.49276$. Only correct to four digits, not five.)
(Edited (2): Gerry Myerson, in the comments, helpfully pointed out a reference that contained the product I was looking at, though starting at 3 rather than 5. It turns out that my original product converges to $12C_2e^{-2\gamma} \approx 2.497287$, which just happens to be very very close to $4\lambda \approx 2.497320$. The $0.0013%$ difference between the makes them easy enough to confuse when you don't know what you're approaching asymptotically.)
I wanted to follow up on this briefly, in case anyone was actually watching it. I found the reasoning for the constant given in Finch. First, we take all of the infinite products involved and start them at 3--rather than at 2 (Merten's 3rd theorem) or 5 (the product I was working with). H&L's $C_2$ already starts at 3. Then we have these, assuming $ 2 < p < n $ for all products:
$$C_2 = \prod\left({\frac{p(p-2)}{(p-1)^2}}\right) =\prod\left({\frac{p}{p-1}}\right) \cdot\prod\left({\frac{p-2}{p-1}}\right) \tag 1$$ $$\frac{2e^{-\gamma}}{\ln n} \sim \prod\left({\frac{p-1}{p}}\right) \tag 2$$
Then multiplying $(1)$ by $(2)$ (leaving behind equality but keeping asymptotic notation):
$$C_2\frac{2e^{-\gamma}}{\ln n} \sim \prod\left({\frac{p-2}{p-1}}\right) $$
If we multiply by $(2)$ a second time, we get:
$$C_2\frac{4e^{-2\gamma}}{(\ln n)^2} \sim \prod\left({\frac{p-2}{p-1}}\right) \cdot\prod\left({\frac{p-1}{p}}\right) = \prod\left({\frac{p-2}{p}}\right) $$
$$C_2\frac{4e^{-2\gamma}}{(\ln n)^2} \sim \prod\left({1-\frac2p}\right)$$ which is the result given by Finch, and the result I was looking for in the original question.