Cramér's conjecture

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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

[math]\displaystyle{ p_{n+1}-p_n=O((\log p_n)^2),\ }[/math]

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

[math]\displaystyle{ \limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1, }[/math]

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.

Conditional proven results on prime gaps

Cramér gave a conditional proof of the much weaker statement that

[math]\displaystyle{ p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n) }[/math]

on the assumption of the Riemann hypothesis.[1] The best known unconditional bound is

[math]\displaystyle{ p_{n+1}-p_n = O(p_n^{0.525}) }[/math]

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]

[math]\displaystyle{ \limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty. }[/math]

His result was improved by R. A. Rankin,[4] who proved that

[math]\displaystyle{ \limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}\cdot\frac{\left(\log\log\log p_n\right)^{2}}{ \log\log p_n \log\log\log\log p_n} \gt 0. }[/math]

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.[6] The two sets of authors improved the result by a [math]\displaystyle{ \log \log \log p_n }[/math] factor later that year.[7]

Heuristic justification

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,

[math]\displaystyle{ \limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{\log^2 p_n} = 1 }[/math]

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 [math]\displaystyle{ c \ge 2e^{-\gamma}\approx1.1229\ldots }[/math] (OEIS: A125313), where [math]\displaystyle{ \gamma }[/math] 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 [math]\displaystyle{ c\gt 2 }[/math], there is a constant [math]\displaystyle{ d\gt 0 }[/math] such that there is a prime between [math]\displaystyle{ x }[/math] and [math]\displaystyle{ x+d(\log x)^c }[/math]. [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

Prime gap function

Daniel Shanks conjectured the following asymptotic equality, stronger than Cramér's conjecture,[13] for record gaps: [math]\displaystyle{ G(x)\sim \log^2 x. }[/math]

J.H. Cadwell[14] has proposed the formula for the maximal gaps: [math]\displaystyle{ G(x)\sim \log^2 x - \log x\log\log x, }[/math] 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 [math]\displaystyle{ G(x) }[/math] expressed in terms of the prime-counting function [math]\displaystyle{ \pi(x) }[/math]:

[math]\displaystyle{ G(x)\sim \frac{x}{\pi(x)}(2\log\pi(x)-\log x+c), }[/math]

where [math]\displaystyle{ c=\log(C_2)=0.2778769... }[/math] and [math]\displaystyle{ C_2=1.3203236... }[/math] 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

[math]\displaystyle{ G(x) \sim \log^2 x - 2\log x\log\log x - (1-c)\log x. }[/math].

Thomas Nicely has calculated many large prime gaps.[16] He measures the quality of fit to Cramér's conjecture by measuring the ratio

[math]\displaystyle{ R = \frac{\log p_n}{\sqrt{p_{n+1}-p_n}}. }[/math]

He writes, "For the largest known maximal gaps, [math]\displaystyle{ R }[/math] has remained near 1.13."

See also

References

  1. ↑ 1.0 1.1 1.2 Cramér, Harald (1936), "On the order of magnitude of the difference between consecutive prime numbers", Acta Arithmetica 2: 23–46, doi:10.4064/aa-2-1-23-46, http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf, retrieved 2012-03-12 
  2. ↑ The Difference Between Consecutive Primes, II, Wiley, 2001, doi:10.1112/plms/83.3.532 
  3. ↑ Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind" (in de), Commentationes Physico-Mathematicae Helsingsfors 5: 1–37 .
  4. ↑ R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247
  5. ↑ Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016). "Large gaps between consecutive prime numbers". Annals of Mathematics. Second series 183 (3): 935–974. doi:10.4007/annals.2016.183.3.4. 
  6. ↑ Maynard, James (2016). "Large gaps between primes". Annals of Mathematics. Second series 183 (3): 915–933. doi:10.4007/annals.2016.183.3.3. 
  7. ↑ Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2018). "Long gaps between primes". Journal of the American Mathematical Society 31: 65–105. doi:10.1090/jams/876. 
  8. ↑ Terry Tao, 254A, Supplement 4: Probabilistic models and heuristics for the primes (optional), section on The Cramér random model, January 2015.
  9. ↑ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers", Scandinavian Actuarial Journal 1: 12–28, doi:10.1080/03461238.1995.10413946, http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf, retrieved 2007-06-05 .
  10. ↑ János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63:2 (April 1997), pp. 286–301.
  11. ↑ 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.
  12. ↑ Robin Visser, Large Gaps Between Primes, University of Cambridge (2020).
  13. ↑ Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation (American Mathematical Society) 18 (88): 646–651, doi:10.2307/2002951 .
  14. ↑ Cadwell, J. H. (1971), "Large Intervals Between Consecutive Primes", Mathematics of Computation 25 (116): 909–913, doi:10.2307/2004355 
  15. ↑ Wolf, Marek (2014), "Nearest-neighbor-spacing distribution of prime numbers and quantum chaos", Phys. Rev. E 89 (2): 022922, doi:10.1103/physreve.89.022922, PMID 25353560, Bibcode: 2014PhRvE..89b2922W, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.89.022922 
  16. ↑ Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation 68 (227): 1311–1315, doi:10.1090/S0025-5718-99-01065-0, Bibcode: 1999MaCom..68.1311N .

External links



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