# the $n$th twin prime, asymptotically

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.)

• 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 $\pi_2(t_n)=n$. Commented Oct 12, 2021 at 8:50
• 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? Commented Oct 12, 2021 at 8:53
• See this post. Commented Oct 12, 2021 at 9:25
• 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.
– Gary
Commented Oct 12, 2021 at 10:34
• @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.
– Gary
Commented Oct 13, 2021 at 23:29

An asymptotic formulae is given for the number of prime pairs $$(p, p+2) is given in the paper, "Asymptotic formulae for the number of repeating prime sequences less than N", Christopher L. Garvie (2016), Notes on Number Theory and Discrete Mathematics, Vol. 22, No. 4; 29-40.
The paper solves the more general problem noted in Hardy and Wight (1962) "An introduction to the Theory of Numbers" on page 5 concerning the number of prime pairs, prime triplets $$(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.