2

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.)

Jesse Elliott
  • 859
  • 1
    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 $\pi_2(t_n)=n$. – Dietrich Burde Oct 12 '21 at 08:50
  • 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? – Jesse Elliott Oct 12 '21 at 08:53
  • See this post. – Dietrich Burde Oct 12 '21 at 09:25
  • 2
    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 Oct 12 '21 at 10:34
  • Nice, if I knew how to invert asymptotics! I suppose you take the series inversion of its generating function as a formal power series? But I'm really after the converse. I suppose if you "invert" the inverted you get back to where you started? – Jesse Elliott Oct 13 '21 at 08:45
  • @DietrichBurde: I mentioned that I already know how to prove the forward direction, which is what you posted. I'm not sure how to prove the converse. Take the series inversion, is what I'm gathering from Gary. – Jesse Elliott Oct 13 '21 at 08:51
  • I thought the answer by Nilotpal Sinha above shows that $\pi(x)\sim x/\log(x)$ implies $p_n\sim n\log(n)$. Isn't this what you wanted, only now for $t_n$? – Dietrich Burde Oct 13 '21 at 08:54
  • No, I want the converse too. Actually, I'm more interested in the converse, since the forward direction appears to be fairly simple. I hear people say all the time that $p_n \sim n \log n$ is equivalent to the PNT, but I'm unable to locate a proof of the forward direction or supply a proof. Same goes for $t_n$. I assume maybe the proof will be similar. – Jesse Elliott Oct 13 '21 at 13:17
  • I tried to convey this in the latter part of the question: "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?" – Jesse Elliott Oct 13 '21 at 13:20
  • 1
    @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 https://doi.org/10.1006/jsco.1994.1014 see Corollary 4 for the case of the prime numbers. – Gary Oct 13 '21 at 23:29
  • Beautiful! I had not seen that asymptotic expansion before in connection to the Hardy-Littlewood conjecture. Is there a reference for that too? – Jesse Elliott Oct 15 '21 at 11:13
  • So $$\frac{\pi_2(x)}{\pi(x)} - 2C_2 \frac{\pi(x)}{x} = o((\log x)^{-n})$$ for all $n$? I had not seen this strengthening of the Hardy-Littlewood conjecture. – Jesse Elliott Oct 15 '21 at 11:18
  • You just observe that the integral is $\operatorname{Li}(x) -\frac{x}{\log x}$ and use the asymptotic series of $\operatorname{Li}(x)$. Please use @Gary otherwise I am not notified about your questions. – Gary Oct 20 '21 at 01:48
  • Just because they are asymptotic doesn't mean they share the same asymptotic series. Your conjecture is much stronger than any version I've seen of the Hardy--Littlewood conjecture. – Jesse Elliott Oct 20 '21 at 01:50
  • That is correct. If one compares the actual values of $\pi_2(x)$ and the integral approximation, it seems that about half of the digits are correct, hence the error is likely beyond all orders. (similarly to the case of $\pi(x)$) – Gary Oct 20 '21 at 01:55
  • https://math.stackexchange.com/questions/4299011/a-formula-that-counts-exactly-the-twin-prime-pairs-occuring-in-an-interval-a-b – Daniel Donnelly Nov 07 '21 at 21:15

1 Answers1

0

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.

Chris G
  • 1