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.)

  • 1
    $\begingroup$ 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
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ @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

1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .