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Are there any known special properties of a number located between twin primes?

The question came up in the discussion. (The expression below has been rephrased to a weaker form for clarity)

In studying admissibility of k-tuples I observed that

$$\varphi( \lambda ) \le \lim_{\lambda < n \le \infty} \inf \varphi( n ) $$

where we use the set theoretical representation of a twin prime pair as $\{\lambda-1, \lambda+1\} \subset \mathbb{P}$ and the definition of $\lim \inf x_n$ is given by $\lim_{N\rightarrow \infty} \inf x_k: k \in [N, \infty]$, and I'm not quite sure how to prove the statement above on the totient of a "lumenal" (?) number beyond conjecture. I think amWhy is curious as well.

This post may be related, because this is a special case of the same problem. Now, of course, with the explicit formula for Euler totient, $$ \varphi(n) = n \prod_{p|n} \bigg( 1 - \frac{1}{p} \bigg) $$ we do, in fact, end up with $\frac{ \varphi(n)}{n}$ having the product form in terms of the prime divisors of $n$ giving the value $n-1$ when $n \in \mathbb{P}$. If we were not prime, then there would be another factor less than one in the formula for totient. Similar logic applies for the number $n+1$, because $n+1$ cannot be prime and as a result $\varphi(n+1) < n$. But this doesn't entirely explain the conjectured expression.

For context, we arrive at a similar expression for the totient of a prime number $$\varphi( p ) > \sup_{0 < n < p} \varphi( n ) $$

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    $\begingroup$ I don't think I understand what you're saying. Can you give an example of what you mean? $\endgroup$ – Carl Schildkraut Jul 16 '17 at 2:40
  • $\begingroup$ Is there supposed to be a liminf on both sides of the equality? $\endgroup$ – mathworker21 Jul 16 '17 at 2:54
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    $\begingroup$ there is no lim inf,as $\varphi(n) \geq \sqrt {\frac{n}{2}}$ see math.stackexchange.com/questions/301837/… $\endgroup$ – Will Jagy Jul 16 '17 at 3:29
  • $\begingroup$ @WillJagy And you can do much better with the primorials and the Mertens theorems to obtain $\inf_{n \le k}\frac{\varphi(n)}{n} \sim \frac{e^{- \gamma}}{\log \log k}$. On the other hand the number of twin primes $\le x$ has a conjectured (but unproven) asymptotics $\pi_2(x) \sim C \frac{x}{(\log x)^2}$ $\endgroup$ – reuns Jul 16 '17 at 3:36
  • $\begingroup$ user1329514 Could you explain why you think the link at the very bottom of your post is related? It could very well be, but it would be helpful for you to articulate how the post there is related to your question. $\endgroup$ – Namaste Jul 16 '17 at 16:36

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