查看“︁哈代-李特爾伍德第一猜想”︁的源代码

{{Infobox mathematical statement
| name = First Hardy–Littlewood conjecture
| image = Anzahl Primzahl-Zwillingspaare.png
| caption = 小於給定的<math>n</math>的[[孿生質數]]的數量的圖。哈代-李特爾伍德第一猜想預測說會有無限多對這樣的數。
| field = [[數論]]
| conjectured by = [[G·H·哈代]]<br>[[約翰·恩瑟·李特爾伍德]]
| conjecture date = 1923
| open problem = 是
| first proof by = 
| first proof date = 
| implied by = 
}}
在[[數論]]中，'''哈代-李特爾伍德第一猜想'''（first Hardy–Littlewood conjecture）{{sfn|Aletheia-Zomlefer|Fukshansky|Garcia|2020}}指的是對小於給定數的{{link-en|質數k元組|Prime_k-tuple}}的非病態公式，這猜想是對[[質數定理]]的推廣。這猜想最初由[[G·H·哈代]]和[[約翰·恩瑟·李特爾伍德]]在1923年提出。<ref name="original">{{cite journal |last1=Hardy |first1=G. H. |author-link1=G. H. Hardy |last2=Littlewood |first2=J. E. |author-link2=John Edensor Littlewood |title=Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes. |journal=[[Acta Mathematica|Acta Math.]] |issue=44 |pages=1–70 |year=1923 |volume=44|doi=10.1007/BF02403921|doi-access=free }}.</ref>

==陳述==
設<math>m_1, m_2, \ldots, m_k </math>為一組使得<math>P = (p, p + m_1, p + m_2, \ldots , p + m_k)</math>不對任何質數構成一個完全剩餘系的正整數，並以<math>\pi_{P}(n)</math>表示不大於<math>n</math>並使得<math>p + m_1, p + m_2, \ldots , p + m_k</math>皆為質數的質數<math>p</math>的數量，那麼有：{{sfn|Aletheia-Zomlefer|Fukshansky|Garcia|2020}}{{sfn|Tóth|2019}}
:<math>\pi_P(n)\sim C_P\int_2^n \frac{dt}{\log^{k+1}t},</math>
其中
:<math>C_P=2^k \prod_{q \text{ prime,} \atop q \ge 3}\frac{1-\frac{w(q; m_1, m_2, \ldots , m_k)}q}{\left(1-\frac{1}{q}\right)^{k+1}}</math>
是奇質數的乘積，且此處<math>w(q; m_1, m_2, \ldots , m_k)</math>表示<math>0, m_1, m_2, \ldots , m_k</math>除以<math>q</math>後，其中不同的餘數的個數。

<math>k=1</math>且<math>m_1=2</math>的情況和[[孿生質數猜想]]相關，特別地，若以<math>\pi_2(n)</math>表示不大於<math>n</math>的孿生質數個數，那麼有
:<math>\pi_2(n)\sim C_2 \int_2^n \frac{dt}{\log^2 t},</math>

其中
:<math>C_2 = 2\prod_{\textstyle{q \text{ prime,}\atop q \ge 3}} \left(1 - \frac{1}{(q-1)^2} \right) \approx  1.320323632\ldots</math>

是孿生質數常數。{{sfn|Tóth|2019}}

==斯奎斯數==
{{main article|質數k元組#斯奎斯數}}
質數k元組的斯奎斯數，是根據哈代-李特爾伍德第一猜想，在質數k元組上對斯奎斯數的定義的推廣。質數k元組<math>P</math>的斯奎斯數的定義是最小的違反哈代-李特爾伍德的質數<math>p</math>，也就是最小的使得下式成立的質數：{{sfn|Tóth|2019}}
:<math> \pi_P(p)>C_P \operatorname {li}_P(p), </math>

==結果==
目前已證明說，哈代-李特爾伍德第一猜想和[[哈代-李特尔伍德第二猜想]]彼此不相容。<ref>{{cite journal | first=Ian | last=Richards | title=On the Incompatibility of Two Conjectures Concerning Primes | journal=Bull. Amer. Math. Soc. | volume=80 | pages=419–438 | year=1974 | doi=10.1090/S0002-9904-1974-13434-8 | doi-access=free }}</ref>

==推廣==
{{link-en|Bateman–Horn猜想|Bateman–Horn conjecture}}是哈代-李特爾伍德第一猜想在次數大於一的多項式上的推廣。{{sfn|Aletheia-Zomlefer|Fukshansky|Garcia|2020}}

==出處==
{{reflist}}

==參考資料==
* {{cite journal |title = The Bateman–Horn conjecture: Heuristic, history, and applications |journal = Expositiones Mathematicae |volume = 38 |number = 4 |pages = 430–479 |year = 2020 |issn = 0723-0869 |doi = 10.1016/j.exmath.2019.04.005 |first1 = Soren Laing |last1 = Aletheia-Zomlefer  |first2 = Lenny |last2 = Fukshansky |first3 = Stephan Ramon |last3 = Garcia|doi-access = free }}
* {{cite journal |last = Tóth |first = László |date = January 2019 |pages = 143–138 |title = On the Asymptotic Density of Prime k-tuples and a Conjecture of Hardy and Littlewood |volume = 25 |journal = Computational Methods in Science and Technology |issue = 3 |doi = 10.12921/cmst.2019.0000033|arxiv = 1910.02636 }}

{{質數猜想}}
{{DEFAULTSORT:First Hardy-Littlewood conjecture}}
[[Category:素數猜想]]