1
$\begingroup$

By the PNT the average gap is approximately the size of the logarithm of one of the primes. However, on one hand Cràmer conjectured $$ p_{n+1}-p_n<\log^2p_n$$and on the other hand there's the twin prime conjecture.

Let $$A_m=\left\lvert\{n\le m: p_{n+1}-p_n>\log p_n\}\right\rvert$$ and $$B_m=\left\lvert\{n\le m: p_{n+1}-p_n<\log p_n\}\right\rvert.$$ Is $\liminf\limits_{m\to\infty} \frac{A_m}{B_m}$ known?

  • 0
    The prime gap is frequently quite large due to the fact that all numbers between $n!+2$ and $n!+n$ are composite. Here $n!$ can be replaced with something close to $e^n$, i.e. $\text{lcm}(2,3,\ldots,n)$. – 2017-02-04
  • 0
    @JackD'Aurizio: Yes. However $n<\log n!$ for $n\ge6$, so I don't think that helps on the long run, right? – 2017-02-04
  • 0
    However, the improved counter-example with $\text{lcm}(2,3,\ldots,n)$ shows that $p_{n+1}-p_{n}\geq C\log(p_n)$ with $C\approx 1$ infinitely often. – 2017-02-04
  • 0
    I would add that by the work of Goldston and Yildirim it is known (http://mathworld.wolfram.com/PrimeGaps.html) that $$\liminf_{n\to +\infty}\;\frac{p_{n+1}-p_{n}}{\log p_n}=0$$ that also follows from recent breakthroughs (Zhang, Maynard, Tao) in the prime gap problem (https://terrytao.wordpress.com/tag/prime-gaps/) – 2017-02-04
  • 1
    @JackD'Aurizio: Yep, thanks for the information. To expand on that, from Wikipedia: "A Polymath Project collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013. In November 2013, James Maynard introduced a new refinement of the GPY sieve, allowing him to reduce the bound to 600 and show that for any m there exists a bounded interval containing m prime numbers. Using Maynard's ideas, the Polymath project improved the bound to 246; assuming the Elliott–Halberstam conjecture and its generalized form, N has been reduced to 12 and 6, respectively." – 2017-02-05
  • 1
    At the bottom of the same page it is mentioned that Ford, Green, Konyagin, Maynard, Tao proved $$p_{n+1}-p_n>\frac{\log p_n \log\log p_n}{\log\log\log p_n}\log\log\log p_n $$ infinitely often. – 2017-02-05

1 Answers 1

1

Some exploration on primes up to $10$ million:

gaps $q-p>\ln(q)$: $253517 $
gaps $q-p<\ln(q)$: $411061 $

The graph of $\frac{q-p}{\ln q}$ against $q$ is a little misleading because the smaller gaps are all concentrated into distinct value lines so appear to take up less space on the graph:

enter image description here

  • 0
    So $A_n/B_n$ seems to stabilise between $1/2$ and $1$. I guess it might be hard to show it – 2017-02-05