Cramér's conjecture

In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936,[1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they can be. It states that

$p_{n+1}-p_n=O((\log p_n)^2),\$

where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement

$\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1,$

and this formulation is often called Cramér's conjecture in the literature.

Neither form of Cramér's conjecture has yet been proven or disproven.

Conditional proven results on prime gaps

Cramér also gave a conditional proof of the much weaker statement that

$p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n)$

on the assumption of the Riemann hypothesis.[1]

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,[2]

$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.$

Heuristic justification

Cramér's conjecture is based on a probabilistic model (essentially a heuristic) of the primes, in which one assumes that the probability of a natural number of size x being prime is 1/log x. This is known as the Cramér model of the primes. Cramér proved that in this model, the above conjecture holds true with probability one.[1]

Heuristics of Shanks and Cramér–Granville conjectures

Prime gap function

Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture.[3]

In the random model,

$\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = c,$ with $c = 1.$

But this constant, $c$, may not apply to all the primes, by Maier's theorem. As pointed out by Andrew Granville,[4] a refinement of Cramér's model taking into account divisibility by small primes suggests that $c \ge 2e^{-\gamma}\approx1.1229\ldots$, where $\gamma$ is the Euler–Mascheroni constant.

Thomas Nicely has calculated many large prime gaps.[5] He measures the quality of fit to Cramér's conjecture by measuring the ratio

$R = \frac{\log(p_n)}{\sqrt{p_{n+1}-p_n}}.$

He writes, “For the largest known maximal gaps, $R$ has remained near 1.13.” However, $1/R^2$ is still less than 1, and it does not provide support to Granville's refinement that c should be greater than 1.