Cramér's conjecture

2017-06-23T17:12:18Z

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In [[number theory]], '''Cramér's conjecture''', formulated by the Swedish mathematician [[Harald Cramér]] in 1936,<ref name="Cramér1936">{{Citation |last=Cramér |first=Harald |title=On the order of magnitude of the difference between consecutive prime numbers |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf |journal=[[Acta Arithmetica]] |volume=2 |year=1936 |pages=23–46 }}</ref> is an estimate for the size of [[prime gap|gaps between consecutive prime numbers]]: intuitively, that gaps between consecutive primes are always small, and the [[conjecture]] quantifies [[asymptotics|asymptotically]] just how small they must be. It states that
:<math>p_{n+1}-p_n=O((\log p_n)^2),\ </math>
where ''p''''n'' denotes the ''n''th [[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
:<math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1,</math>
and this formulation is often called Cramér's conjecture in the literature. However, this stronger formulation is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture.

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

==Conditional proven results on prime gaps==
Cramér gave a [[conditional proof]] of the much [[List of mathematical jargon#weak|weaker]] statement that
:<math>p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n)</math>
on the assumption of the [[Riemann hypothesis]].<ref name="Cramér1936" /> The best known unconditional bound is
:<math>p_{n+1}-p_n = O(p_n^{0.525})</math>
due to Baker, Harman, and Pintze.<ref>R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83 (2001), no. 3, 532-562</ref>

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,<ref>{{Citation |last=Westzynthius |first=E. |title=Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind |language=de|journal=Commentationes Physico-Mathematicae Helsingsfors |volume=5 |issue= |year=1931 |pages=1–37 | zbl=0003.24601 | jfm=57.0186.02 }}.</ref>
:<math>\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.</math>
His result was improved by [[R. A. Rankin]],<ref>R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247</ref> who proved that
:<math>\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n \log\log p_n \log^{-2} \log\log p_n \log\log\log\log p_n} > 0.</math>
[[Paul Erdős]] conjectured that the left-hand side of the above formula is equal to infinity, and this was proven in 2014 by [[Kevin Ford (mathematician)|Kevin Ford]], [[Ben Green (mathematician)|Ben Green]], [[Sergei Konyagin]], and [[Terence Tao]].<ref>K. Ford, B. Green, S. Konyagin, and T. Tao, Large gaps between consecutive prime numbers. Ann. of Math. (2) 183 (2016), no. 3, 935–974</ref>

==Heuristic justification==
Cramér's conjecture is based on a [[Probabilistic number theory|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.

In the Cramér random model,
:<math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{\log^2 p_n} = 1</math>
with [[Almost sure convergence|probability one]].<ref name="Cramér1936" /> However, as pointed out by [[Andrew Granville]],<ref>{{Citation |last=Granville |first=A. |title=Harald Cramér and the distribution of prime numbers |journal=Scandinavian Actuarial Journal |volume=1 |issue= |year=1995 |pages=12–28 |url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf |doi=10.1080/03461238.1995.10413946}}.</ref> [[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>c \ge 2e^{-\gamma}\approx1.1229\ldots</math> ({{OEIS2C|id=A125313}}), where <math>\gamma</math> is the [[Euler–Mascheroni constant]]. János Pintz has suggested that the [[Limit superior and limit inferior|limit sup]] may be infinite,<ref>János Pintz, Very large gaps between consecutive primes, ''Journal of Number Theory'' '''63''':2 (April 1997), pp. 286–301.</ref> 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>c>2</math>, there is a constant <math>d>0</math> such that there is a prime between <math>x</math> and <math>x+d(\log x)^c</math>.'' <ref>[[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.</ref>

==Related conjectures and heuristics==
[[File:Wikipedia primegaps.png|thumb|294px|Prime gap function]]
[[Daniel Shanks]] conjectured the following asymptotic equality for record gaps:
<math>G(x)\sim \log^2 x.</math>
This is a stronger statement than Cramér's conjecture.<ref>{{Citation |first=Daniel |last=Shanks |title=On Maximal Gaps between Successive Primes |journal=Mathematics of Computation |volume=18 |issue=88 |year=1964 |pages=646–651 |doi=10.2307/2002951 |publisher=American Mathematical Society |jstor=2002951|zbl=0128.04203 }}.</ref>

In the paper <ref name="Cadwell1971">{{Citation |last=Cadwell |first= J. H. |title=Large Intervals Between Consecutive Primes |journal= Mathematics of Computation |volume= 25 |issue=116 |year=1971 |pages=909–913 |jstor=2004355 |doi=10.2307/2004355}}</ref> J.H. Cadwell has proposed the formula for the maximal gaps:
<math>G(x)\sim \log x(\log x-\log\log x),</math>
which for large <math> x</math> implies both the Cramér and Shanks conjectures.

[[Thomas Nicely#Chronology|Thomas Nicely]] has calculated many large prime gaps.<ref>{{Citation |last=Nicely |first=Thomas R. |doi=10.1090/S0025-5718-99-01065-0 |mr=1627813 |issue=227 |journal=Mathematics of Computation |pages=1311–1315 |title=New maximal prime gaps and first occurrences |url=http://www.trnicely.net/gaps/gaps.html |volume=68 |year=1999 }}.</ref> He measures the quality of fit to Cramér's conjecture by measuring the ratio
:<math>R = \frac{\log p_n}{\sqrt{p_{n+1}-p_n}}.</math>

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

==See also==
*[[Prime number theorem]]
*[[Legendre's conjecture]] and [[Andrica's conjecture]], much weaker but still unproven upper bounds on prime gaps
* [[Firoozbakht's conjecture]]
* [[Maier's theorem]] on the numbers of primes in short intervals for which the model predicts an incorrect answer

==References==
{{Reflist}}
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 | isbn=978-0-387-20860-2 | zbl=1058.11001 | at=A8 }}
* {{cite journal | authorlink=János Pintz | last1=Pintz | first1=János | title=Cramér vs. Cramér. On Cramér's probabilistic model for primes | url=http://projecteuclid.org/euclid.facm/1229619660 | mr=2363833 | year=2007 | journal= Functiones et Approximatio Commentarii Mathematici | volume=37 | pages=361–376 | zbl=1226.11096 | issn=0208-6573 | doi=10.7169/facm/1229619660}}
* {{cite book | last=Soundararajan | first=K. | authorlink=Kannan Soundararajan | chapter=The distribution of prime numbers | editor1-last=Granville | editor1-first=Andrew | editor1-link=Andrew Granville | editor2-last=Rudnick | editor2-first=Zeév | title=Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005 | location=Dordrecht |publisher=[[Springer-Verlag]] | series=NATO Science Series II: Mathematics, Physics and Chemistry | volume=237 | pages=59–83 | year=2007 | isbn=978-1-4020-5403-7 | zbl=1141.11043 }}

==External links==
*{{mathworld|title=Cramér Conjecture|urlname=CramerConjecture}}
*{{mathworld|title=Cramér-Granville Conjecture|urlname=Cramer-GranvilleConjecture}}

{{Prime number conjectures}}

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[[Category:Conjectures about prime numbers]]

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Cramér's conjecture