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Assuming the first Hardy--Littlewood conjecture for twin primes, what is an asymptotic formula for the $n$th twin prime $t_n$ (which, let us say, enumerates the first of the each twin prime pair, so that the sequence $t_n$ is $3,5,11,17,29,\ldots$)? I imagine it will take the form $$t_n \sim C n \,(\log n)^2 \ (n \to \infty)$$ for some constant $C$ expressible in terms of the twin prime constant. Also, is the Hardy--Littlewood conjecture for twin primes equivalent to such an asymptotic expression for the $n$th twin prime $t_n$? If so, how would you deduce the Hardy--Littlewood conjecture for twin primes from such an asymptotic relation for $t_n$?

A more basic question is, what is the easiest way to deduce the prime number theorem from the asymptotic relation $$p_n \sim n \log n \ (n \to \infty),$$ where $p_n$ denotes the $n$th prime? Maybe a proof in the twin prime case would be similar?

(Pardon the notation $t_n$ for the $n$th twin prime. If you have a better notation, let me know and I will edit the question.)

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    $\begingroup$ The PNT $\pi(x)\sim \frac{x}{\log(x)}$ is equivalent to $p_n\sim n\log(n)$ using $\pi(p_n)=n$. Similarly $$ \pi_2(x)\sim 2C_2 \frac{x}{\log (x) ^2}, $$ for $C_2=0.66016...$ gives an asymptotc for $t_n$, using $\pi_2(t_n)=n$. $\endgroup$ Commented Oct 12, 2021 at 8:50
  • $\begingroup$ I know how to prove the forward direction for the PNT, but how do you prove the reverse direction? And what does the asymptotic for $t_n$ turn out to be? $\endgroup$ Commented Oct 12, 2021 at 8:53
  • $\begingroup$ See this post. $\endgroup$ Commented Oct 12, 2021 at 9:25
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    $\begingroup$ If you invert the conjectured asymptotics $$ \pi _2 (x) \sim 2C_2 \int_2^x {\frac{{dt}}{{\log ^2 t}}} $$ then you find $$ t_n \sim \frac{1}{{2C_2 }}n\log ^2 n + 2C_2 (n\log n)\log \log n + \frac{{1 - \log (2C_2 )}}{{C_2 }}n\log n + \cdots $$ Here $C_2 = 0.6601618158468\ldots$ is the twin prime constant. $\endgroup$
    – Gary
    Commented Oct 12, 2021 at 10:34
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    $\begingroup$ @JesseElliott You start with the asymptotic expansion $$ \pi _2 (x) \sim 2C_2 \int_2^x {\frac{{dt}}{{\log ^2 t}} \sim } 2C_2 \frac{x}{{\log ^2 x}}\sum\limits_{k = 0}^\infty {\frac{{(k + 1)!}}{{\log ^k x}}} $$ and use Theorem 2 in doi.org/10.1006/jsco.1994.1014 see Corollary 4 for the case of the prime numbers. $\endgroup$
    – Gary
    Commented Oct 13, 2021 at 23:29

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An asymptotic formulae is given for the number of prime pairs $(p, p+2) <N$ is given in the paper, "Asymptotic formulae for the number of repeating prime sequences less than N", Christopher L. Garvie (2016), Notes on Number Theory and Discrete Mathematics, Vol. 22, No. 4; 29-40.

The paper solves the more general problem noted in Hardy and Wight (1962) "An introduction to the Theory of Numbers" on page 5 concerning the number of prime pairs, prime triplets $(p,p+2,p+6)$ & $(p,p+4,p=6)$, and also larger sets of primes. A sieve method is used in the paper to show all such sequences (including the prime pair conjecture) are unbounded.

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