克拉梅尔猜想

本文使用了数学技术上的对数表记。在不另外说明的状况下，本文中所有的

\log {x}

都应视为自然对数，也就是一般常记为

\ln {x}

或

\log _{e}{x}

的对数。

数学上的克拉梅尔猜想（Cramér's conjecture）是瑞典数学家哈拉尔德·克拉梅尔在1937年提出的关于质数间隙的猜想。^[1]该猜想是说：

\limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{(\ln p_{n})^{2}}}=1

，

这里 $p_{n}$ 代表第 $n$ 个素数。该猜想到现在仍未证出或被否证。

关于质数间隙的条件结果

克拉梅尔也提出另一个较弱的关于素数间隙的猜想，指出在黎曼猜想成立的状况下，有

p_{n+1}-p_{n}=O({\sqrt {p_{n}}}\,\ln p_{n})

。^[1]

目前这方面最好的无条件结果是

p_{n+1}-p_{n}=O(p_{n}^{0.525})

而这点由R·C·贝克（R. C. Baker）、格林·哈曼（英语：Glyn Harman）和平茨·亚诺什（匈牙利语：Pintz János）三人证出。^[2]

另一方面，E·韦斯钦蒂乌斯（E. Westzynthius）于1931年证明质数间隙成长速度快过对数，也就是说，^[3]

\limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=\infty .

罗伯特·亚历山大·兰金（英语：Robert Alexander Rankin）改进了他的结果，^[4]并证明道

\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}}}>0

艾狄胥·帕尔猜想表示上式的左侧趋近于无限，而这点于2014年由凯文·福特（英语：Kevin Ford (mathematician)）、本·格林（英语：Ben Green (mathematician)）、谢尔盖·科尼亚金（英语：Sergei Konyagin）和陶哲轩四人组。^[5]以及詹姆斯·梅纳德分别证出。^[6]这两组人马在该年稍晚将该结果以 $\log \log \log p_{n}$ 因子进行改进。^[7]

探索性论证

克拉梅尔猜想是基于本质上探索性的机率模型（英语：Probabilistic number theory）之上的，在其中一个大小为 $x$ 的数是质数的机率是 $1/\log {x}$ 。而该结果又称作“克拉梅尔随机模型”（Cramér random model）或“克拉梅尔质数模型”（Cramér model of the primes）。^[8]

根据克拉梅尔随机模型，以下事件的机率为一^[1]：

\limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{\log ^{2}p_{n}}}=1

然而，安德鲁·格兰维尔（英语：Andrew Granville）指出，^[9]根据迈尔定理，克拉梅尔随机模型不能适切地描述质数在短区间上的分布，而在考虑可除性后，修正版克拉梅尔模型指向 $c\geq 2e^{-\gamma }\approx 1.1229\ldots$ （A125313），其中 $\gamma$ 是欧拉-马斯刻若尼常数。平茨·亚诺什则认为该比值的上极限可能发散至无限；^[10]

类似地，伦纳德·阿德曼和凯文·麦柯利（Kevin McCurley）写道：

“由于H. Maier关于相邻质数间隙的工作之故，学界对克拉梅尔猜想的确实公式起了疑问…（中略）因此很有可能对于任意的常数

c>2

而言，总存在一个常数，使得

x

和

x+d(\log x)^{c}

有一个质数。”^[11]

类似地，罗宾·维瑟（Robin Visser）写道：

“事实上，由于格兰维尔的工作之故，现在学界普遍相信克拉梅尔猜想是错的。实际上也确实有迈尔定理等关于短区间的定理，和克拉梅尔模型难以兼容。”^[12]

参见

质数定理
勒让德猜想和安德里卡猜想─两个远远较弱但依旧未证出的关于质数间隙上界的猜想。
菲鲁兹巴赫特猜想
迈尔定理─一个关于短区间质数个数且克拉梅尔模型给出错误预测的定理。

参考资料

^ ^1.0 ^1.1 ^1.2 Cramér, Harald, On the order of magnitude of the difference between consecutive prime numbers (PDF), Acta Arithmetica, 1936, 2: 23–46 [2012-03-12], doi:10.4064/aa-2-1-23-46, （原始内容 (PDF)存档于2018-07-23）
^ 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
^ Westzynthius, E., Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind, Commentationes Physico-Mathematicae Helsingsfors, 1931, 5: 1–37, JFM 57.0186.02, Zbl 0003.24601 （德语） .
^ R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247
^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence. Large gaps between consecutive prime numbers. Annals of Mathematics. Second series. 2016, 183 (3): 935–974. arXiv:1408.4505 . doi:10.4007/annals.2016.183.3.4 .
^ Maynard, James. Large gaps between primes. Annals of Mathematics. Second series. 2016, 183 (3): 915–933. arXiv:1408.5110 . doi:10.4007/annals.2016.183.3.3 .
^ Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence. Long gaps between primes. Journal of the American Mathematical Society. 2018, 31: 65–105. arXiv:1412.5029 . doi:10.1090/jams/876.
^ Terry Tao, 254A, Supplement 4: Probabilistic models and heuristics for the primes (optional), section on The Cramér random model, January 2015.
^ Granville, A., Harald Cramér and the distribution of prime numbers (PDF), Scandinavian Actuarial Journal, 1995, 1: 12–28 [2007-06-05], doi:10.1080/03461238.1995.10413946, （原始内容 (PDF)存档于2015-09-23） .
^ János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63:2 (April 1997), pp. 286–301.
^ Leonard Adleman（英语：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.
^ Robin Visser, Large Gaps Between Primes, University of Cambridge (2020).
^ Shanks, Daniel, On Maximal Gaps between Successive Primes, Mathematics of Computation (American Mathematical Society), 1964, 18 (88): 646–651, JSTOR 2002951, Zbl 0128.04203, doi:10.2307/2002951  .
^ Cadwell, J. H., Large Intervals Between Consecutive Primes, Mathematics of Computation, 1971, 25 (116): 909–913, JSTOR 2004355, doi:10.2307/2004355 
^ Wolf, Marek, Nearest-neighbor-spacing distribution of prime numbers and quantum chaos, Phys. Rev. E, 2014, 89 (2): 022922, Bibcode:2014PhRvE..89b2922W, PMID 25353560, S2CID 25003349, arXiv:1212.3841 , doi:10.1103/physreve.89.022922
^ Nicely, Thomas R., New maximal prime gaps and first occurrences, Mathematics of Computation, 1999, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, MR 1627813, doi:10.1090/S0025-5718-99-01065-0  .

Guy, Richard K. Unsolved problems in number theory 3rd. Springer-Verlag. 2004. A8. ISBN 978-0-387-20860-2. Zbl 1058.11001.
Pintz, János. Cramér vs. Cramér. On Cramér's probabilistic model for primes. Functiones et Approximatio Commentarii Mathematici. 2007, 37 (2): 361–376. ISSN 0208-6573. MR 2363833. Zbl 1226.11096. doi:10.7169/facm/1229619660 .
Soundararajan, K. The distribution of prime numbers. Granville, Andrew; Rudnick, Zeév (编). 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 237. Dordrecht: Springer-Verlag. 2007: 59–83. ISBN 978-1-4020-5403-7. Zbl 1141.11043.

外部链接

[Cramér1936-1] 1.0 ^1.1 ^1.2 Cramér, Harald, On the order of magnitude of the difference between consecutive prime numbers (PDF), Acta Arithmetica, 1936, 2: 23–46 [2012-03-12], doi:10.4064/aa-2-1-23-46, （原始内容 (PDF)存档于2018-07-23）

[2] 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

[3] Westzynthius, E., Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind, Commentationes Physico-Mathematicae Helsingsfors, 1931, 5: 1–37, JFM 57.0186.02, Zbl 0003.24601 （德语） .

[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. Large gaps between consecutive prime numbers. Annals of Mathematics. Second series. 2016, 183 (3): 935–974. arXiv:1408.4505 . doi:10.4007/annals.2016.183.3.4 .

[6] Maynard, James. Large gaps between primes. Annals of Mathematics. Second series. 2016, 183 (3): 915–933. arXiv:1408.5110 . doi:10.4007/annals.2016.183.3.3 .

[7] Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence. Long gaps between primes. Journal of the American Mathematical Society. 2018, 31: 65–105. arXiv:1412.5029 . 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., Harald Cramér and the distribution of prime numbers (PDF), Scandinavian Actuarial Journal, 1995, 1: 12–28 [2007-06-05], doi:10.1080/03461238.1995.10413946, （原始内容 (PDF)存档于2015-09-23） .

[10] János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63:2 (April 1997), pp. 286–301.

[11] Leonard Adleman（英语：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, On Maximal Gaps between Successive Primes, Mathematics of Computation (American Mathematical Society), 1964, 18 (88): 646–651, JSTOR 2002951, Zbl 0128.04203, doi:10.2307/2002951  .

[Cadwell1971-14] Cadwell, J. H., Large Intervals Between Consecutive Primes, Mathematics of Computation, 1971, 25 (116): 909–913, JSTOR 2004355, doi:10.2307/2004355 

[Wolf2014-15] Wolf, Marek, Nearest-neighbor-spacing distribution of prime numbers and quantum chaos, Phys. Rev. E, 2014, 89 (2): 022922, Bibcode:2014PhRvE..89b2922W, PMID 25353560, S2CID 25003349, arXiv:1212.3841 , doi:10.1103/physreve.89.022922

[16] Nicely, Thomas R., New maximal prime gaps and first occurrences, Mathematics of Computation, 1999, 68 (227): 1311–1315, Bibcode:1999MaCom..68.1311N, MR 1627813, doi:10.1090/S0025-5718-99-01065-0  .

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关于质数间隙的条件结果

探索性论证

相关猜想和探索

参见

参考资料

外部链接