查看“︁林德勒夫猜想”︁的源代码

'''林德勒夫猜想'''（Lindelöf hypothesis）是一個由[[芬蘭]]數學家{{link-en|恩斯特·雷納德·林德勒夫|Ernst Leonard Lindelöf}}提出一個關於[[黎曼ζ函數]]在臨界線上增長率的猜想。<ref>參見{{harvtxt|Lindelöf|1908}}</ref>這猜想可由[[黎曼猜想]]導出，其形式以[[大O符號]]表述如下：

對於任意的<math>\varepsilon > 0</math>而言，在<math>t</math>趨近於無窮時，有<math>\zeta\!\left(\frac{1}{2} + it\right)\! = O(t^\varepsilon)</math>

由於<math>\varepsilon</math>可由一個較小的值取代之故，因此這猜想可重述如下：

對於任意的<math>\varepsilon > 0</math>而言，有<math>\zeta\!\left(\frac{1}{2} + it\right)\! = o(t^\varepsilon)</math>

==μ函數==
設<math>\sigma</math>是一個[[實數]]，則可定義<math>\mu(\sigma)</math>為所有使得<math>\zeta(\sigma+iT)=O(T^a)</math>的實數<math>a</math>當中的最小數。在這種定義下，易見對於任意的<math>\sigma > 1</math>，有<math>\mu(\sigma)=0</math>，而從黎曼ζ函數的[[函數方程]]可導出說<math>\mu(\sigma)=\mu(1-\sigma)-\sigma+\frac{1}{2}</math>。另一方面，由{{link-en|夫拉門–林德勒夫定理|Phragmén–Lindelöf theorem}}可導出說<math>\mu</math>是一個[[凸函數]]。林德勒夫猜想基本就是說，<math>\mu(\frac{1}{2})=0</math>，將此點和上述的性質結合，這猜想也意味著說在<math>\sigma \ge \frac{1}{2}</math>時，<math>\mu(\sigma)=0</math>；而在<math>\sigma \le \frac{1}{2}</math>時，<math>\mu(\sigma)=\frac{1}{2}-\sigma</math>

由於<math>\mu(1)=0</math>且<math>\mu(0)=\frac{1}{2}</math>，因此從林德勒夫對這函數的凸性可導出說<math>0\le\mu(\frac{1}{2})=\frac{1}{4}</math>。之後[[G·H·哈代]]藉由將[[赫爾曼·外爾|外爾]]估計{{link-en|指數和|Exponential sum}}的方式用於[[黎曼-西格爾公式|近似函數方程]]的做法，將這上界降至<math>\frac{1}{6}</math>。在那之後數名研究者用長且技術性的[[數學證明]]，將之降到稍微低於<math>\frac{1}{6}</math>的數值。下表顯示了對於這數值的改進：
{| class="wikitable"
|-
! ''μ''(1/2)&nbsp;≤
! ''μ''(1/2)&nbsp;≤
! 研究者
|-
| 1/4
| 0.25
|Lindelöf<ref>{{harvtxt|Lindelöf|1908}}</ref>
|凸性上界
|-
| 1/6
| 0.1667
|Hardy & Littlewood<ref name="Hardy & Littlewood 1923 pp. 403–412">{{cite journal
	| last1 = Hardy 
	| first1 = G. H. 
	| last2 = Littlewood
	| first2 = J. E.

	| date = 1923
	| title = On Lindelöf's hypothesis concerning the Riemann zeta-function
	| journal = Proc. R. Soc. A
	| pages = 403–412
}}</ref><ref name="Hardy Littlewood 1916 pp. 119–196">{{cite journal | last1=Hardy | first1=G. H. | last2=Littlewood | first2=J. E. | title=Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes | journal=Acta Mathematica | volume=41 | date=1916 | issn=0001-5962 | doi=10.1007/BF02422942 | pages=119–196}}</ref>
|-
|163/988
|0.1650
|Walfisz 1924<ref name="Walfisz 1924 pp. 115-143">{{cite journal
	| last1 = Walfisz 
	| first1 = Arnold 
	| date = 1924
	| title = Zur Abschätzung von ζ(½ + it)
	| journal = Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse
	| pages = 155–158
}}</ref>
|-
|27/164
|0.1647
|Titchmarsh 1932<ref name="Titchmarsh 1932 pp. 133–141">{{cite journal | last=Titchmarsh | first=E. C. | title=On van der Corput's method and the zeta-function of Riemann (III) | journal=The Quarterly Journal of Mathematics | volume=os-3 | issue=1 | date=1932 | issn=0033-5606 | doi=10.1093/qmath/os-3.1.133 | pages=133–141}}</ref>
|-
|229/1392
|0.164512
|Phillips 1933<ref name="Phillips 1933 pp. 209–225">{{cite journal | last=Phillips | first=Eric | title=The zeta-function of Riemann: further developments of van der Corput's method | journal=The Quarterly Journal of Mathematics | volume=os-4 | issue=1 | date=1933 | issn=0033-5606 | doi=10.1093/qmath/os-4.1.209 | pages=209–225}}</ref>
|-
|
|0.164511
|Rankin 1955<ref name="Rankin 1955 pp. 147–153">{{cite journal | last=Rankin | first=R. A. | title=Van der Corput's method and the theory of exponent pairs | journal=The Quarterly Journal of Mathematics | volume=6 | issue=1 | date=1955 | issn=0033-5606 | doi=10.1093/qmath/6.1.147 | pages=147–153}}</ref>
|-
|19/116
|0.1638
|Titchmarsh 1942<ref name="Titchmarsh 1942 pp. 11–17">{{cite journal | last=Titchmarsh | first=E. C. | title=On the order of ζ(½+ it ) | journal=The Quarterly Journal of Mathematics | volume=os-13 | issue=1 | date=1942 | issn=0033-5606 | doi=10.1093/qmath/os-13.1.11 | pages=11–17}}</ref>
|-
|15/92
|0.1631
|Min 1949<ref name="Min 1949 pp. 448–472">{{cite journal | last=Min | first=Szu-Hoa | title=On the order of 𝜁(1/2+𝑖𝑡) | journal=Transactions of the American Mathematical Society | volume=65 | issue=3 | date=1949 | issn=0002-9947 | doi=10.1090/S0002-9947-1949-0030996-6 | pages=448–472}}</ref>
|-
|6/37
|0.16217
|Haneke 1962<ref name="Haneke 1963 pp. 357–430">{{cite journal | last=Haneke | first=W. | title=Verschärfung der Abschätzung von ξ(½+it) | journal=Acta Arithmetica | volume=8 | issue=4 | date=1963 | language=German | issn=0065-1036 | doi=10.4064/aa-8-4-357-430 | pages=357–430}}</ref>
|-
|173/1067
|0.16214
|Kolesnik 1973<ref name="Kolesnik 1973 pp. 7-30">{{cite journal | last=Kolesnik | first=G. A. | title=On the estimation of some trigonometric sums | journal=Acta Arithmetica | volume=25 | issue=1 |language=Russian | year=1973 | issn=0065-1036 | pages=7–30 | url=https://eudml.org/doc/205256 | access-date=2024-02-05}}</ref>
|-
|35/216
|0.16204
|Kolesnik 1982<ref name="Kolesnik 1982 pp. 107–122">{{cite journal | last=Kolesnik | first=Grigori | title=On the order of ζ (1/2+ it ) and Δ( R ) | journal=Pacific Journal of Mathematics | volume=98 | issue=1 | date=1982-01-01 | issn=0030-8730 | doi=10.2140/pjm.1982.98.107 | pages=107–122}}</ref>
|-
|139/858
|0.16201
|Kolesnik 1985<ref name="Kolesnik 1985 pp. 115-143">{{cite journal
	| last1 = Kolesnik
	| first1 = G. 
	| date = 1985
	| title = On the method of exponent pairs
	| journal = Acta Arithmetica
	| volume = 45
	| issue = 2
	| pages = 115–143
| doi = 10.4064/aa-45-2-115-143 
	}}</ref>
|-
|9/56
|0.1608
|Bombieri & Iwaniec 1986<ref>{{cite journal |last1=Bombieri |first1=E. |last2=Iwaniec |first2=H. | title=On the order of ζ (1/2+ it ) |journal=Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |date=1986 |volume=13 |issue=3 |pages=449–472}}</ref>
|-
|32/205
|0.1561
|Huxley<ref>{{harvtxt|Huxley|2002}}, {{harvtxt|Huxley|2005}}</ref>
|-
|53/342
|0.1550
|Bourgain<ref>{{harvtxt|Bourgain|2017}}</ref>
|-
|13/84
|0.1548
|Bourgain<ref>{{harvtxt|Bourgain|2017}}</ref>
|}

==和黎曼猜想間的關係==
Backlund<ref>{{harvtxt|Backlund|1918–1919}}</ref>在1918至1919年間，證明了說林德勒夫猜想和下述與黎曼ζ函數的[[零點]]相關的敘述等價：在<math>T</math>趨近於無窮時，[[實部]]至少為<math>\frac{1}{2}+\varepsilon</math>且[[虛部]]介於<math>T</math>和<math>T+1</math>之間的零點，其數量會趨近於<math>o(\log{T})</math>。

由於黎曼猜想指稱在這區域中沒有任何零點之故，因此黎曼猜想會導出林德勒夫猜想。目前已知[[虛部]]介於<math>T</math>和<math>T+1</math>之間的零點的數量為<math>O(\log{T})</math>，因此林德勒夫猜想似乎只稍強於已知的結果，但盡管如此，人們迄今依舊無法證明林德勒夫猜想。

==黎曼ζ函數的冪的平均值==
林德勒夫猜想與以下陳述等價：

對於任意的正[[整數]]<math>k</math>和正實數<math>\varepsilon</math>而言，有以下等式：

:<math>\frac{1}{T} \int_0^T|\zeta(1/2+it)|^{2k}\,dt = O(T^{\varepsilon})</math>

目前已證明這等式對<math>k=1</math>及<math>k=2</math>成立，但<math>k=3</math>的情況似乎困難許多，且依舊是個[[未解決的問題]]。

對於這[[積分]]的非病態行為，有著下列更加精確的猜想：

一般認為，對某些常數<math>c_{k,j}</math>而言，有以下等式：

:<math> \int_0^T|\zeta(1/2+it)|^{2k} \, dt = T\sum_{j=0}^{k^2}c_{k,j}\log(T)^{k^2-j} + o(T)</math>

李特爾伍德證明了<math>k=1</math>的情況，而希斯-布朗<ref>{{harvtxt|Heath-Brown|1979}}</ref>藉由推廣英厄姆（Ingham）找到首項係數的結果<ref>{{harvtxt|Ingham|1928}}</ref>，證明了<math>k=2</math>的情況。

Conrey和Ghosh<ref>{{harvtxt|Conrey|Ghosh|1998}}</ref>推測，在<math>k=6</math>時首項係數應當為
:<math>\frac{42}{9!}\prod_ p \left((1-p^{-1})^4(1+4p^{-1}+p^{-2})\right)</math> 

而Keating和Snaith<ref>{{harvtxt|Keating|Snaith|2000}}</ref>利用[[隨機矩陣]]理論，對<math>k</math>更大的情況的係數的值做出了一些猜測。目前猜想這積分的首項係數的值是某個初等因子、[[質數]]的某種乘積，和由下列[[數列]]給出的<math>n\times n</math>[[楊表]]的數字彼此間的乘積：
:1, 1, 2, 42, 24024, 701149020, ... {{OEIS|id=A039622}}

==其他後果==
設<math>p_n</math>為第<math>n</math>個[[質數]]，並設<math>g_n = p_{n + 1} - p_n.\ </math>為[[質數間隙]]，則一個由{{link-en|阿爾伯特·英厄姆|Albert Ingham}}證明的結果顯示，若林德勒夫猜想成立，則對於任意的<math>\varepsilon > 0</math>而言，當<math>n</math>{{link-en|最終地 (數學)|Eventually (mathematics)|足夠大}}時，有以下不等式：

:<math>g_n\ll p_n^{1/2+\varepsilon}</math>

對於質數間隙，一個比英厄姆的結果更強的猜想是[[克拉梅爾猜想]]，其陳述如下：<ref name="Cramér1936">{{cite journal |last=Cramér |first=Harald |date=1936 |title=On the order of magnitude of the difference between consecutive prime numbers |journal=Acta Arithmetica |volume=2 |issue=1 |pages=23–46 |doi=10.4064/aa-2-1-23-46 |issn=0065-1036}}</ref><ref name="Banks Ford Tao 2023 pp. 1471–1518">{{cite journal |last1=Banks |first1=William |last2=Ford |first2=Kevin |last3=Tao |first3=Terence |date=2023 |title=Large prime gaps and probabilistic models |journal=Inventiones Mathematicae |volume=233 |issue=3 |pages=1471–1518 |arxiv=1908.08613 |doi=10.1007/s00222-023-01199-0 |issn=0020-9910}}</ref> 
:<math> g_n = O\!\left((\log p_n)^2\right).</math>

===密度假說===
[[File:DensityHypothesis.png|thumb|已知的無零點區域，略合於此張圖的右下角；而若黎曼猜想得證，就會將整張圖給壓縮到x軸上，也就是<math>A_{RH}(\sigma>1/2)=0</math>。在另一邊，此圖中的上界<math>A_{DH}(1-\sigma)=2(1-1/2)=1</math>與從{{link-en|黎曼-馮·曼戈爾特公式|Riemann–von Mangoldt formula}}得出的顯著上界相合。（也有其他各式各樣的估計<ref>{{Cite arXiv |last1=Trudgian |first1=Timothy S. |last2=Yang |first2=Andrew |date=2023 |title=Toward optimal exponent pairs |class=math.NT |eprint=2306.05599}}</ref>）]]

密度假說指稱<math>N(\sigma,T)\le N^{2(1-\sigma)+\varepsilon}</math>，其中<math>N(\sigma,T)</math>是<math>\zeta(s)</math>的零點<math>\rho</math>在<math>\mathfrak{R}(s)\ge \sigma</math>以及<math>|\mathfrak{I}(s)|\le T</math>所構成的範圍內的數量，且這假說可由林德勒夫猜想得出。<ref>{{Cite web |title=25a |url=https://aimath.org/WWN/rh/articles/html/25a/ |access-date=2024-07-16 |website=aimath.org}}</ref><ref>{{Cite web |title=Density hypothesis - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Density_hypothesis |access-date=2024-07-16 |website=encyclopediaofmath.org}}</ref>

更一般地，設<math>N(\sigma,T)\le N^{A(\sigma)(1-\sigma)+\varepsilon}</math>，則已知這界限大致和長度為<math>x^{1-1/A(\sigma)}</math>的短區間當中的質數的漸進公式相合。<ref>{{Cite web |date=2024-06-04 |title=New Bounds for Large Values of Dirichlet Polynomials, Part 1 - Videos {{!}} Institute for Advanced Study |url=https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-1 |access-date=2024-07-16 |website=www.ias.edu |language=en}}</ref><ref>{{Cite web |date=2024-06-04 |title=New Bounds for Large Values of Dirichlet Polynomials, Part 2 - Videos {{!}} Institute for Advanced Study |url=https://www.ias.edu/video/new-bounds-large-values-dirichlet-polynomials-part-2 |access-date=2024-07-16 |website=www.ias.edu |language=en}}</ref>

{{link-en|阿爾伯特·英厄姆|Albert Ingham|英厄姆}}在1940年證明說<math>A_I(\sigma)=\frac{3}{2-\sigma}</math>，<ref>{{Cite journal |last=Ingham |first=A. E. |date=1940 |title=ON THE ESTIMATION OF N (σ, T ) |url=https://academic.oup.com/qjmath/article-lookup/doi/10.1093/qmath/os-11.1.201 |journal=The Quarterly Journal of Mathematics |language=en |volume=os-11 |issue=1 |pages=201–202 |doi=10.1093/qmath/os-11.1.201 |issn=0033-5606}}</ref>{{link-en|馬丁·赫胥黎|Martin Huxley|赫胥黎}}在1971年證明說<math>A_H(\sigma)=\frac{3}{3\sigma-1}</math>；<ref>{{Cite journal |last=Huxley |first=M. N. |date=1971 |title=On the Difference between Consecutive Primes. |url=https://eudml.org/doc/142126 |journal=Inventiones Mathematicae |volume=15 |issue=2 |pages=164–170 |doi=10.1007/BF01418933 |issn=0020-9910}}</ref>
而{{link-en|拉里·古斯|Larry Guth|古斯}}及[[詹姆斯·梅納德|梅納德]]在2024年的一篇預印本中證明說<math>A_{GM}(\sigma)=\frac{15}{5\sigma+3}</math><ref>{{Cite arXiv |last1=Guth |first1=Larry |last2=Maynard |first2=James |date=2024 |title=New large value estimates for Dirichlet polynomials |class=math.NT |eprint=2405.20552}}</ref><ref>{{Cite web |last=Bischoff |first=Manon |title=The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved |url=https://www.scientificamerican.com/article/the-riemann-hypothesis-the-biggest-problem-in-mathematics-is-a-step-closer/ |access-date=2024-07-16 |website=Scientific American |language=en}}</ref><ref>{{Cite web |last=Cepelewicz |first=Jordana |date=2024-07-15 |title='Sensational' Proof Delivers New Insights Into Prime Numbers |url=https://www.quantamagazine.org/sensational-proof-delivers-new-insights-into-prime-numbers-20240715/ |access-date=2024-07-16 |website=Quanta Magazine |language=en}}</ref>並證明說這些公式和<math>\sigma_{I,GM}=7/10<\sigma_{H,GM}=8/10<\sigma_{I,H}=3/4</math>相契合。因此古斯和梅納德近期的成果給出了已知最接近<math>\sigma=1/2</math>、符合一般對黎曼猜想期望的數值，並將其界限改進至<math>N(\sigma,T)\le N^{\frac{30}{13}(1-\sigma)+\varepsilon}</math>，或等價地，非病態地和<math>x^{17/30}</math>成比例。

在理論上，貝克、{{link-en|格林·哈曼|Glyn Harman|哈曼}}和{{link-hu|平茨·亞諾什|Pintz János|平茨}}三氏對[[勒讓德猜想]]的估計的改進、對沒有[[西格爾零點]]的區域的估計，以及其他的事情也是可期待的。

==L函數==
黎曼ζ函數屬於一類被稱為[[L函數]]的一類更加一般的函數。

在2010年，{{link-en|約瑟夫·伯恩斯坦|Joseph Bernstein}}及安德烈·瑞斯妮可夫（Andre Reznikov）給出了估計定義在<math>PGL(2)</math>之上的L函數的次凸性值的方法；<ref>{{Cite journal|last1=Bernstein|first1=Joseph|last2=Reznikov|first2=Andre|date=2010-10-05|title=Subconvexity bounds for triple L -functions and representation theory|url=http://annals.math.princeton.edu/2010/172-3/p05|journal=Annals of Mathematics|language=en|volume=172|issue=3|pages=1679–1718|doi=10.4007/annals.2010.172.1679|s2cid=14745024|issn=0003-486X|doi-access=free|arxiv=math/0608555}}</ref>同一年，[[阿克沙伊·文卡泰什]]及{{link-en|飛利浦·麥可|Philippe Michel (number theorist)}}給出了估計定義在<math>GL(1)</math>和<math>GL(2)</math>之上的L函數的次凸性值的方法；<ref name="SubconvexityGL2">{{cite journal|last1=Michel|first1=Philippe|author-link1=Philippe Michel (number theorist)|last2=Venkatesh|first2=Akshay|year=2010|title=The subconvexity problem for GL<sub>2</sub>|journal=[[Publications Mathématiques de l'IHÉS]]|volume=111|issue=1|pages=171–271|arxiv=0903.3591|citeseerx=10.1.1.750.8950|doi=10.1007/s10240-010-0025-8|s2cid=14155294}}</ref>而在2021年，保羅·尼爾森（Paul Nelson）估計定義在<math>GL(n)</math>之上的L函數的值的方法。<ref>{{cite arXiv|last=Nelson|first=Paul D.|date=2021-09-30|title=Bounds for standard $L$-functions|class=math.NT|eprint=2109.15230}}</ref><ref>{{Cite web|last=Hartnett|first=Kevin|date=2022-01-13|title=Mathematicians Clear Hurdle in Quest to Decode Primes|url=https://www.quantamagazine.org/mathematicians-clear-hurdle-in-quest-to-decode-prime-numbers-20220113/|access-date=2022-02-17|website=Quanta Magazine|language=en}}</ref>

== 參見 ==

* {{link-en|Z函數|Z function}}中的林德勒夫猜想

==註解和參考資料==

{{Reflist}}
*{{citation|last=Backlund|first= R. |title=Über die Beziehung zwischen Anwachsen und Nullstellen der Zeta-Funktion|journal= Ofversigt Finska Vetensk. Soc. |volume=61|issue=9|year=1918–1919|url=https://www.biodiversitylibrary.org/item/51075#page/359/mode/1up}}
*{{Citation | last1=Bourgain | first1=Jean | authorlink=Jean Bourgain | title=Decoupling, exponential sums and the Riemann zeta function | journal=[[Journal of the American Mathematical Society]] | doi=10.1090/jams/860 |arxiv=1408.5794 | year=2017 | volume=30 | pages=205–224 | mr=3556291 | issue=1| s2cid=118064221 }}
*{{Citation | last1=Conrey | first1=J. B. | last2=Farmer | first2=D. W. | last3=Keating | first3=Jonathan P. | last4=Rubinstein | first4=M. O. | last5=Snaith | first5=N. C. | title=Integral moments of L-functions | doi=10.1112/S0024611504015175 | mr=2149530 | year=2005 | journal=Proceedings of the London Mathematical Society |series=Third Series | issn=0024-6115 | volume=91 | issue=1 | pages=33–104| arxiv=math/0206018 | s2cid=1435033 }}
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*{{Citation | last1=Keating | first1=Jonathan P. | last2=Snaith | first2=N. C. | title=Random matrix theory and ζ(1/2+it) | doi=10.1007/s002200000261 | mr=1794265 | year=2000 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=214 | issue=1 | pages=57–89| bibcode=2000CMaPh.214...57K | citeseerx=10.1.1.15.8362 | s2cid=11095649 }}
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*{{eom|id=L/l058960|first=S.M.|last= Voronin}}

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