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. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven.
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. This is known as the Cramér random model or Cramér model of the primes.[8]
In the Cramér random model,
with probability one.[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 (OEIS: A125313), where is the Euler–Mascheroni constant. János Pintz has suggested that the limit sup may be infinite,[10] and similarly Leonard Adleman and Kevin McCurley write
As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cramér's conjecture has been called into question [...] It is still probably true that for every constant , there is a constant such that there is a prime between and .[11]
Similarly, Robin Visser writes
In fact, due to the work done by Granville, it is now widely believed that Cramér's conjecture is false. Indeed, there some theorems concerning short intervals between primes, such as Maier's theorem, which contradict Cramér's model.[12]
(internal references removed).
Related conjectures and heuristics
Daniel Shanks conjectured the following asymptotic equality, stronger than Cramér's conjecture,[13] for record gaps:
J.H. Cadwell[14] has proposed the formula for the maximal gaps:
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
expressed in terms of the prime-counting function:
where and is twice the twin primes constant; see OEIS: A005597, OEIS: A114907. This is again formally equivalent to the Shanks conjecture but suggests lower-order terms
.
Thomas Nicely has calculated many large prime gaps.[16] He measures the quality of fit to Cramér's conjecture by measuring the ratio
He writes, "For the largest known maximal gaps, has remained near 1.13."
^Baker, R. C., Harman, G., Pintz, J. (2001), The Difference Between Consecutive Primes, II, Wiley, doi:10.1112/plms/83.3.532
^Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM57.0186.02, Zbl0003.24601.
^R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247
^János Pintz, Very large gaps between consecutive primes, Journal of Number Theory63:2 (April 1997), pp. 286–301.
^Leonard Adleman and Kevin McCurley, Open Problems in Number Theoretic Complexity, II. Algorithmic number theory (Ithaca, NY, 1994), 291–322, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.
Soundararajan, K. (2007). "The distribution of prime numbers". In Granville, Andrew; Rudnick, Zeév (eds.). Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht: Springer-Verlag. pp. 59–83. ISBN978-1-4020-5403-7. Zbl1141.11043.