林德勒夫猜想

林德勒夫猜想（Lindelöf hypothesis）是一個由芬蘭數學家恩斯特·雷納德·林德勒夫（英语：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。^[1]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 $\varepsilon >0$ 而言，在 $t$ 趨近於無窮時，有 $\zeta \!\left({\frac {1}{2}}+it\right)\!=O(t^{\varepsilon })$

由於 $\varepsilon$ 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 $\varepsilon >0$ 而言，有 $\zeta \!\left({\frac {1}{2}}+it\right)\!=o(t^{\varepsilon })$

μ函數

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

由於 $\mu (1)=0$ 且 $\mu (0)={\frac {1}{2}}$ ，因此從林德勒夫對這函數的凸性可導出說 $0\leq \mu ({\frac {1}{2}})={\frac {1}{4}}$ 。之後G·H·哈代藉由將外爾估計指數和（英语：Exponential sum）的方式用於近似函數方程的做法，將這上界降至 ${\frac {1}{6}}$ 。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 ${\frac {1}{6}}$ 的數值。下表顯示了對於這數值的改進：

μ(1/2) ≤	μ(1/2) ≤	研究者
1/4	0.25	Lindelöf^[2]	凸性上界
1/6	0.1667	Hardy & Littlewood^[3]^[4]
163/988	0.1650	Walfisz 1924^[5]
27/164	0.1647	Titchmarsh 1932^[6]
229/1392	0.164512	Phillips 1933^[7]
	0.164511	Rankin 1955^[8]
19/116	0.1638	Titchmarsh 1942^[9]
15/92	0.1631	Min 1949^[10]
6/37	0.16217	Haneke 1962^[11]
173/1067	0.16214	Kolesnik 1973^[12]
35/216	0.16204	Kolesnik 1982^[13]
139/858	0.16201	Kolesnik 1985^[14]
9/56	0.1608	Bombieri & Iwaniec 1986^[15]
32/205	0.1561	Huxley^[16]
53/342	0.1550	Bourgain^[17]
13/84	0.1548	Bourgain^[18]

和黎曼猜想間的關係

Backlund^[19]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 $T$ 趨近於無窮時，實部至少為 ${\frac {1}{2}}+\varepsilon$ 且虛部介於 $T$ 和 $T+1$ 之間的零點，其數量會趨近於 $o(\log {T})$ 。

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 $T$ 和 $T+1$ 之間的零點的數量為 $O(\log {T})$ ，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 $k$ 和正實數 $\varepsilon$ 而言，有以下等式：

{\frac {1}{T}}\int _{0}^{T}|\zeta (1/2+it)|^{2k}\,dt=O(T^{\varepsilon })

目前已證明這等式對 $k=1$ 及 $k=2$ 成立，但 $k=3$ 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 $c_{k,j}$ 而言，有以下等式：

\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)

李特爾伍德證明了 $k=1$ 的情況，而希斯-布朗^[20]藉由推廣英厄姆（Ingham）找到首項係數的結果^[21]，證明了 $k=2$ 的情況。

Conrey和Ghosh^[22]推測，在 $k=6$ 時首項係數應當為

{\frac {42}{9!}}\prod _{p}\left((1-p^{-1})^{4}(1+4p^{-1}+p^{-2})\right)

而Keating和Snaith^[23]利用隨機矩陣理論，對 $k$ 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 $n\times n$ 楊表的數字彼此間的乘積：

1, 1, 2, 42, 24024, 701149020, ... （OEIS數列A039622）

其他後果

設 $p_{n}$ 為第 $n$ 個質數，並設 $g_{n}=p_{n+1}-p_{n}.\$ 為質數間隙，則一個由阿爾伯特·英厄姆（英语：Albert Ingham）證明的結果顯示，若林德勒夫猜想成立，則對於任意的 $\varepsilon >0$ 而言，當 $n$ 足夠大（英语：Eventually (mathematics)）時，有以下不等式：

g_{n}\ll p_{n}^{1/2+\varepsilon }

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：^[24]^[25]

g_{n}=O\!\left((\log p_{n})^{2}\right).

密度假說

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 $A_{RH}(\sigma >1/2)=0$ 。在另一邊，此圖中的上界 $A_{DH}(1-\sigma )=2(1-1/2)=1$ 與從黎曼-馮·曼戈爾特公式（英语：Riemann–von Mangoldt formula）得出的顯著上界相合。（也有其他各式各樣的估計^[26]）

密度假說指稱 $N(\sigma ,T)\leq N^{2(1-\sigma )+\varepsilon }$ ，其中 $N(\sigma ,T)$ 是 $\zeta (s)$ 的零點 $\rho$ 在 ${\mathfrak {R}}(s)\geq \sigma$ 以及 $|{\mathfrak {I}}(s)|\leq T$ 所構成的範圍內的數量，且這假說可由林德勒夫猜想得出。^[27]^[28]

更一般地，設 $N(\sigma ,T)\leq N^{A(\sigma )(1-\sigma )+\varepsilon }$ ，則已知這界限大致和長度為 $x^{1-1/A(\sigma )}$ 的短區間當中的質數的漸進公式相合。^[29]^[30]

英厄姆（英语：Albert Ingham）在1940年證明說 $A_{I}(\sigma )={\frac {3}{2-\sigma }}$ ，^[31]赫胥黎（英语：Martin Huxley）在1971年證明說 $A_{H}(\sigma )={\frac {3}{3\sigma -1}}$ ；^[32] 而古斯（英语：Larry Guth）及梅納德在2024年的一篇預印本中證明說 $A_{GM}(\sigma )={\frac {15}{5\sigma +3}}$ ^[33]^[34]^[35]並證明說這些公式和 $\sigma _{I,GM}=7/10<\sigma _{H,GM}=8/10<\sigma _{I,H}=3/4$ 相契合。因此古斯和梅納德近期的成果給出了已知最接近 $\sigma =1/2$ 、符合一般對黎曼猜想期望的數值，並將其界限改進至 $N(\sigma ,T)\leq N^{{\frac {30}{13}}(1-\sigma )+\varepsilon }$ ，或等價地，非病態地和 $x^{17/30}$ 成比例。

在理論上，貝克、哈曼（英语：Glyn Harman）和平茨（匈牙利语：Pintz János）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，約瑟夫·伯恩斯坦（英语：Joseph Bernstein）及安德烈·瑞斯妮可夫（Andre Reznikov）給出了估計定義在 $PGL(2)$ 之上的L函數的次凸性值的方法；^[36]同一年，阿克沙伊·文卡泰什及飛利浦·麥可（英语：Philippe Michel (number theorist)）給出了估計定義在 $GL(1)$ 和 $GL(2)$ 之上的L函數的次凸性值的方法；^[37]而在2021年，保羅·尼爾森（Paul Nelson）估計定義在 $GL(n)$ 之上的L函數的值的方法。^[38]^[39]

參見

Z函數（英语：Z function）中的林德勒夫猜想

註解和參考資料

^ 參見Lindelöf (1908)
^ Lindelöf (1908)
^ Hardy, G. H.; Littlewood, J. E. On Lindelöf's hypothesis concerning the Riemann zeta-function. Proc. R. Soc. A. 1923: 403–412.
^ Hardy, G. H.; Littlewood, J. E. Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes. Acta Mathematica. 1916, 41: 119–196. ISSN 0001-5962. doi:10.1007/BF02422942.
^ Walfisz, Arnold. Zur Abschätzung von ζ(½ + it). Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse. 1924: 155–158.
^ Titchmarsh, E. C. On van der Corput's method and the zeta-function of Riemann (III). The Quarterly Journal of Mathematics. 1932, os–3 (1): 133–141. ISSN 0033-5606. doi:10.1093/qmath/os-3.1.133.
^ Phillips, Eric. The zeta-function of Riemann: further developments of van der Corput's method. The Quarterly Journal of Mathematics. 1933, os–4 (1): 209–225. ISSN 0033-5606. doi:10.1093/qmath/os-4.1.209.
^ Rankin, R. A. Van der Corput's method and the theory of exponent pairs. The Quarterly Journal of Mathematics. 1955, 6 (1): 147–153. ISSN 0033-5606. doi:10.1093/qmath/6.1.147.
^ Titchmarsh, E. C. On the order of ζ(½+ it ). The Quarterly Journal of Mathematics. 1942, os–13 (1): 11–17. ISSN 0033-5606. doi:10.1093/qmath/os-13.1.11.
^ Min, Szu-Hoa. On the order of 𝜁(1/2+𝑖𝑡). Transactions of the American Mathematical Society. 1949, 65 (3): 448–472. ISSN 0002-9947. doi:10.1090/S0002-9947-1949-0030996-6.
^ Haneke, W. Verschärfung der Abschätzung von ξ(½+it). Acta Arithmetica. 1963, 8 (4): 357–430. ISSN 0065-1036. doi:10.4064/aa-8-4-357-430 （German）.
^ Kolesnik, G. A. On the estimation of some trigonometric sums. Acta Arithmetica. 1973, 25 (1): 7–30 [2024-02-05]. ISSN 0065-1036 （Russian）.
^ Kolesnik, Grigori. On the order of ζ (1/2+ it ) and Δ( R ). Pacific Journal of Mathematics. 1982-01-01, 98 (1): 107–122. ISSN 0030-8730. doi:10.2140/pjm.1982.98.107.
^ Kolesnik, G. On the method of exponent pairs. Acta Arithmetica. 1985, 45 (2): 115–143. doi:10.4064/aa-45-2-115-143.
^ Bombieri, E.; Iwaniec, H. On the order of ζ (1/2+ it ). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 1986, 13 (3): 449–472.
^ Huxley (2002), Huxley (2005)
^ Bourgain (2017)
^ Bourgain (2017)
^ Backlund (1918–1919)
^ Heath-Brown (1979)
^ Ingham (1928)
^ Conrey & Ghosh (1998)
^ Keating & Snaith (2000)
^ Cramér, Harald. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica. 1936, 2 (1): 23–46. ISSN 0065-1036. doi:10.4064/aa-2-1-23-46.
^ Banks, William; Ford, Kevin; Tao, Terence. Large prime gaps and probabilistic models. Inventiones Mathematicae. 2023, 233 (3): 1471–1518. ISSN 0020-9910. arXiv:1908.08613 . doi:10.1007/s00222-023-01199-0.
^ Trudgian, Timothy S.; Yang, Andrew. Toward optimal exponent pairs. 2023. arXiv:2306.05599  [math.NT].
^ 25a. aimath.org. [2024-07-16].
^ Density hypothesis - Encyclopedia of Mathematics. encyclopediaofmath.org. [2024-07-16].
^ New Bounds for Large Values of Dirichlet Polynomials, Part 1 - Videos | Institute for Advanced Study. www.ias.edu. 2024-06-04 [2024-07-16] （英语）.
^ New Bounds for Large Values of Dirichlet Polynomials, Part 2 - Videos | Institute for Advanced Study. www.ias.edu. 2024-06-04 [2024-07-16] （英语）.
^ Ingham, A. E. ON THE ESTIMATION OF N (σ, T ). The Quarterly Journal of Mathematics. 1940, os–11 (1): 201–202. ISSN 0033-5606. doi:10.1093/qmath/os-11.1.201 （英语）.
^ Huxley, M. N. On the Difference between Consecutive Primes.. Inventiones Mathematicae. 1971, 15 (2): 164–170. ISSN 0020-9910. doi:10.1007/BF01418933.
^ Guth, Larry; Maynard, James. New large value estimates for Dirichlet polynomials. 2024. arXiv:2405.20552  [math.NT].
^ Bischoff, Manon. The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved. Scientific American. [2024-07-16] （英语）.
^ Cepelewicz, Jordana. 'Sensational' Proof Delivers New Insights Into Prime Numbers. Quanta Magazine. 2024-07-15 [2024-07-16] （英语）.
^ Bernstein, Joseph; Reznikov, Andre. Subconvexity bounds for triple L -functions and representation theory. Annals of Mathematics. 2010-10-05, 172 (3): 1679–1718. ISSN 0003-486X. S2CID 14745024. arXiv:math/0608555 . doi:10.4007/annals.2010.172.1679  （英语）.
^ Michel, Philippe; Venkatesh, Akshay. The subconvexity problem for GL₂. Publications Mathématiques de l'IHÉS. 2010, 111 (1): 171–271. CiteSeerX 10.1.1.750.8950 . S2CID 14155294. arXiv:0903.3591 . doi:10.1007/s10240-010-0025-8.
^ Nelson, Paul D. Bounds for standard $L$-functions. 2021-09-30. arXiv:2109.15230  [math.NT].
^ Hartnett, Kevin. Mathematicians Clear Hurdle in Quest to Decode Primes. Quanta Magazine. 2022-01-13 [2022-02-17] （英语）.

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Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C., Integral moments of L-functions, Proceedings of the London Mathematical Society, Third Series, 2005, 91 (1): 33–104, ISSN 0024-6115, MR 2149530, S2CID 1435033, arXiv:math/0206018 , doi:10.1112/S0024611504015175
Conrey, J. B.; Farmer, D. W.; Keating, Jonathan P.; Rubinstein, M. O.; Snaith, N. C., Lower order terms in the full moment conjecture for the Riemann zeta function, Journal of Number Theory, 2008, 128 (6): 1516–1554, ISSN 0022-314X, MR 2419176, S2CID 15922788, arXiv:math/0612843 , doi:10.1016/j.jnt.2007.05.013
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[1] 參見Lindelöf (1908)

[2] Lindelöf (1908)

[Hardy_&_Littlewood_1923_pp._403–412-3] Hardy, G. H.; Littlewood, J. E. On Lindelöf's hypothesis concerning the Riemann zeta-function. Proc. R. Soc. A. 1923: 403–412.

[Hardy_Littlewood_1916_pp._119–196-4] Hardy, G. H.; Littlewood, J. E. Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes. Acta Mathematica. 1916, 41: 119–196. ISSN 0001-5962. doi:10.1007/BF02422942.

[Walfisz_1924_pp._115-143-5] Walfisz, Arnold. Zur Abschätzung von ζ(½ + it). Nachr. Ges. Wiss. Göttingen, math.-phys. Klasse. 1924: 155–158.

[Titchmarsh_1932_pp._133–141-6] Titchmarsh, E. C. On van der Corput's method and the zeta-function of Riemann (III). The Quarterly Journal of Mathematics. 1932, os–3 (1): 133–141. ISSN 0033-5606. doi:10.1093/qmath/os-3.1.133.

[Phillips_1933_pp._209–225-7] Phillips, Eric. The zeta-function of Riemann: further developments of van der Corput's method. The Quarterly Journal of Mathematics. 1933, os–4 (1): 209–225. ISSN 0033-5606. doi:10.1093/qmath/os-4.1.209.

[Rankin_1955_pp._147–153-8] Rankin, R. A. Van der Corput's method and the theory of exponent pairs. The Quarterly Journal of Mathematics. 1955, 6 (1): 147–153. ISSN 0033-5606. doi:10.1093/qmath/6.1.147.

[Titchmarsh_1942_pp._11–17-9] Titchmarsh, E. C. On the order of ζ(½+ it ). The Quarterly Journal of Mathematics. 1942, os–13 (1): 11–17. ISSN 0033-5606. doi:10.1093/qmath/os-13.1.11.

[Min_1949_pp._448–472-10] Min, Szu-Hoa. On the order of 𝜁(1/2+𝑖𝑡). Transactions of the American Mathematical Society. 1949, 65 (3): 448–472. ISSN 0002-9947. doi:10.1090/S0002-9947-1949-0030996-6.

[Haneke_1963_pp._357–430-11] Haneke, W. Verschärfung der Abschätzung von ξ(½+it). Acta Arithmetica. 1963, 8 (4): 357–430. ISSN 0065-1036. doi:10.4064/aa-8-4-357-430 （German）.

[Kolesnik_1973_pp._7-30-12] Kolesnik, G. A. On the estimation of some trigonometric sums. Acta Arithmetica. 1973, 25 (1): 7–30 [2024-02-05]. ISSN 0065-1036 （Russian）.

[Kolesnik_1982_pp._107–122-13] Kolesnik, Grigori. On the order of ζ (1/2+ it ) and Δ( R ). Pacific Journal of Mathematics. 1982-01-01, 98 (1): 107–122. ISSN 0030-8730. doi:10.2140/pjm.1982.98.107.

[Kolesnik_1985_pp._115-143-14] Kolesnik, G. On the method of exponent pairs. Acta Arithmetica. 1985, 45 (2): 115–143. doi:10.4064/aa-45-2-115-143.

[15] Bombieri, E.; Iwaniec, H. On the order of ζ (1/2+ it ). Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. 1986, 13 (3): 449–472.

[16] Huxley (2002), Huxley (2005)

[17] Bourgain (2017)

[18] Bourgain (2017)

[19] Backlund (1918–1919)

[20] Heath-Brown (1979)

[21] Ingham (1928)

[22] Conrey & Ghosh (1998)

[23] Keating & Snaith (2000)

[Cramér1936-24] Cramér, Harald. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithmetica. 1936, 2 (1): 23–46. ISSN 0065-1036. doi:10.4064/aa-2-1-23-46.

[Banks_Ford_Tao_2023_pp._1471–1518-25] Banks, William; Ford, Kevin; Tao, Terence. Large prime gaps and probabilistic models. Inventiones Mathematicae. 2023, 233 (3): 1471–1518. ISSN 0020-9910. arXiv:1908.08613 . doi:10.1007/s00222-023-01199-0.

[26] Trudgian, Timothy S.; Yang, Andrew. Toward optimal exponent pairs. 2023. arXiv:2306.05599  [math.NT].

[27] 25a. aimath.org. [2024-07-16].

[28] Density hypothesis - Encyclopedia of Mathematics. encyclopediaofmath.org. [2024-07-16].

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查论编數論中的L函數
解析例子	黎曼ζ函數狄利克雷L函數赫克特徵的L函數（英语：Hecke character）自守L函數（英语：Automorphic L-function）塞爾伯格類（英语：Selberg class）
代數例子	戴德金ζ函數（英语：Dedekind zeta function）阿廷L函數（英语：Artin L-function）哈斯-韋伊L函數（英语：Hasse–Weil zeta function）動形L函數（英语：Motivic L-function）
定理	類數公式黎曼-馮·曼戈爾特公式（英语：Riemann–von Mangoldt formula）韋伊猜想（英语：Weil conjectures）
解析猜想	黎曼猜想广义黎曼猜想林德勒夫猜想拉馬努金-皮特森猜想（英语：Ramanujan–Petersson conjecture）阿廷L函數猜想（英语：Artin conjecture (L-functions)）
代數猜想	贝赫和斯维讷通-戴尔猜想 L函數的特殊值（英语：Special values of L-functions）朗蘭茲猜想
p進L函數（英语：p-adic L-function）	岩澤理論的主猜想（英语：Main conjecture of Iwasawa theory）塞爾瑪群（英语：Selmer group）歐拉系統（英语：Euler system）