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 must be.

It states that 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 heuristic actually supports the stronger statement and sometimes this formulation is called Cramér's conjecture.

Cramér gave a conditional proof of the much weaker statement that on the assumption of the Riemann hypothesis.

[1] The best known unconditional bound is due to Baker, Harman, and Pintz.

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

That is,[3] His result was improved by R. A. Rankin,[4] who proved that Paul Erdős conjectured that the left-hand side of the above formula is infinite, and this was proven in 2014 by Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao,[5] and independently by James Maynard.

Cramér's conjecture is based on a probabilistic model—essentially a heuristic—in which the probability that a number of size x is prime is 1/log x.

[1] However, as pointed out by Andrew Granville,[9] Maier's theorem shows that the Cramér random model does not adequately describe the distribution of primes on short intervals, and a refinement of Cramér's model taking into account divisibility by small primes suggests that the limit should not be 1, but a constant

János Pintz has suggested that the limit sup may be infinite,[10] and similarly Leonard Adleman and Kevin McCurley write Similarly, Robin Visser writes (internal references removed).

which is formally identical to the Shanks conjecture but suggests a lower-order term.

Marek Wolf[15] has proposed the formula for the maximal gaps

This is again formally equivalent to the Shanks conjecture but suggests lower-order terms Thomas Nicely has calculated many large prime gaps.