林德勒夫猜想

林德勒夫猜想（Lindelöf hypothesis）是一个由芬兰数学家恩斯特·雷纳德·林德勒夫（英语：Ernst Leonard Lindelöf）提出一个关于黎曼ζ函数在临界线上增长率的猜想。^[1]这猜想可由黎曼猜想导出，其形式以大O符号表述如下：

对于任意的 $\varepsilon >0$ 而言，在 $t$ 趋近于无穷时，有 $\zeta \!\left({\frac {1}{2}}+it\right)\!=O(t^{\varepsilon })$

由于 $\varepsilon$ 可由一个较小的值取代之故，因此这猜想可重述如下：

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

μ函数

设 $\sigma$ 是一个实数，则可定义 $\mu (\sigma )$ 为所有使得 $\zeta (\sigma +iT)=O(T^{a})$ 的实数 $a$ 当中的最小数。在这种定义下，易见对于任意的 $\sigma >1$ ，有 $\mu (\sigma )=0$ ，而从黎曼ζ函数的函数方程可导出说 $\mu (\sigma )=\mu (1-\sigma )-\sigma +{\frac {1}{2}}$ 。另一方面，由夫拉门–林德勒夫定理（英语：Phragmén–Lindelöf theorem）可导出说 $\mu$ 是一个凸函数。林德勒夫猜想基本就是说， $\mu ({\frac {1}{2}})=0$ ，将此点和上述的性质结合，这猜想也意味著说在 $\sigma \geq {\frac {1}{2}}$ 时， $\mu (\sigma )=0$ ；而在 $\sigma \leq {\frac {1}{2}}$ 时， $\mu (\sigma )={\frac {1}{2}}-\sigma$

由于 $\mu (1)=0$ 且 $\mu (0)={\frac {1}{2}}$ ，因此从林德勒夫对这函数的凸性可导出说 $0\leq \mu ({\frac {1}{2}})={\frac {1}{4}}$ 。之后G·H·哈代借由将外尔估计指数和（英语：Exponential sum）的方式用于近似函数方程的做法，将这上界降至 ${\frac {1}{6}}$ 。在那之后数名研究者用长且技术性的数学证明，将之降到稍微低于 ${\frac {1}{6}}$ 的数值。下表显示了对于这数值的改进：

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]

和黎曼猜想间的关系

Backlund^[19]在1918至1919年间，证明了说林德勒夫猜想和下述与黎曼ζ函数的零点相关的叙述等价：在 $T$ 趋近于无穷时，实部至少为 ${\frac {1}{2}}+\varepsilon$ 且虚部介于 $T$ 和 $T+1$ 之间的零点，其数量会趋近于 $o(\log {T})$ 。

由于黎曼猜想指称在这区域中没有任何零点之故，因此黎曼猜想会导出林德勒夫猜想。目前已知虚部介于 $T$ 和 $T+1$ 之间的零点的数量为 $O(\log {T})$ ，因此林德勒夫猜想似乎只稍强于已知的结果，但尽管如此，人们迄今依旧无法证明林德勒夫猜想。

黎曼ζ函数的幂的平均值

林德勒夫猜想与以下陈述等价：

对于任意的正整数 $k$ 和正实数 $\varepsilon$ 而言，有以下等式：

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

目前已证明这等式对 $k=1$ 及 $k=2$ 成立，但 $k=3$ 的情况似乎困难许多，且依旧是个未解决的问题。

对于这积分的非病态行为，有著下列更加精确的猜想：

一般认为，对某些常数 $c_{k,j}$ 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)

李特尔伍德证明了 $k=1$ 的情况，而希斯-布朗^[20]借由推广英厄姆（Ingham）找到首项系数的结果^[21]，证明了 $k=2$ 的情况。

Conrey和Ghosh^[22]推测，在 $k=6$ 时首项系数应当为

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

而Keating和Snaith^[23]利用随机矩阵理论，对 $k$ 更大的情况的系数的值做出了一些猜测。目前猜想这积分的首项系数的值是某个初等因子、质数的某种乘积，和由下列数列给出的 $n\times n$ 杨表的数字彼此间的乘积：

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

其他后果

设 $p_{n}$ 为第 $n$ 个质数，并设 $g_{n}=p_{n+1}-p_{n}.\$ 为质数间隙，则一个由阿尔伯特·英厄姆（英语：Albert Ingham）证明的结果显示，若林德勒夫猜想成立，则对于任意的 $\varepsilon >0$ 而言，当 $n$ 足够大（英语：Eventually (mathematics)）时，有以下不等式：

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

对于质数间隙，一个比英厄姆的结果更强的猜想是克拉梅尔猜想，其陈述如下：^[24]^[25]

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

密度假说

已知的无零点区域，略合于此张图的右下角；而若黎曼猜想得证，就会将整张图给压缩到x轴上，也就是 $A_{RH}(\sigma >1/2)=0$ 。在另一边，此图中的上界 $A_{DH}(1-\sigma )=2(1-1/2)=1$ 与从黎曼-冯·曼戈尔特公式（英语：Riemann–von Mangoldt formula）得出的显著上界相合。（也有其他各式各样的估计^[26]）

密度假说指称 $N(\sigma ,T)\leq N^{2(1-\sigma )+\varepsilon }$ ，其中 $N(\sigma ,T)$ 是 $\zeta (s)$ 的零点 $\rho$ 在 ${\mathfrak {R}}(s)\geq \sigma$ 以及 $|{\mathfrak {I}}(s)|\leq T$ 所构成的范围内的数量，且这假说可由林德勒夫猜想得出。^[27]^[28]

更一般地，设 $N(\sigma ,T)\leq N^{A(\sigma )(1-\sigma )+\varepsilon }$ ，则已知这界限大致和长度为 $x^{1-1/A(\sigma )}$ 的短区间当中的质数的渐进公式相合。^[29]^[30]

英厄姆（英语：Albert Ingham）在1940年证明说 $A_{I}(\sigma )={\frac {3}{2-\sigma }}$ ，^[31]赫胥黎（英语：Martin Huxley）在1971年证明说 $A_{H}(\sigma )={\frac {3}{3\sigma -1}}$ ；^[32] 而古斯（英语：Larry Guth）及梅纳德在2024年的一篇预印本中证明说 $A_{GM}(\sigma )={\frac {15}{5\sigma +3}}$ ^[33]^[34]^[35]并证明说这些公式和 $\sigma _{I,GM}=7/10<\sigma _{H,GM}=8/10<\sigma _{I,H}=3/4$ 相契合。因此古斯和梅纳德近期的成果给出了已知最接近 $\sigma =1/2$ 、符合一般对黎曼猜想期望的数值，并将其界限改进至 $N(\sigma ,T)\leq N^{{\frac {30}{13}}(1-\sigma )+\varepsilon }$ ，或等价地，非病态地和 $x^{17/30}$ 成比例。

在理论上，贝克、哈曼（英语：Glyn Harman）和平茨（匈牙利语：Pintz János）三氏对勒让德猜想的估计的改进、对没有西格尔零点的区域的估计，以及其他的事情也是可期待的。

L函数

黎曼ζ函数属于一类被称为L函数的一类更加一般的函数。

在2010年，约瑟夫·伯恩斯坦（英语：Joseph Bernstein）及安德烈·瑞斯妮可夫（Andre Reznikov）给出了估计定义在 $PGL(2)$ 之上的L函数的次凸性值的方法；^[36]同一年，阿克沙伊·文卡泰什及飞利浦·麦可（英语：Philippe Michel (number theorist)）给出了估计定义在 $GL(1)$ 和 $GL(2)$ 之上的L函数的次凸性值的方法；^[37]而在2021年，保罗·尼尔森（Paul Nelson）估计定义在 $GL(n)$ 之上的L函数的值的方法。^[38]^[39]

参见

Z函数（英语：Z function）中的林德勒夫猜想

注解和参考资料

^ 参见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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Ingham, A. E., On the estimation of N(σ,T), The Quarterly Journal of Mathematics, Second Series, 1940, 11 (1): 291–292, Bibcode:1940QJMat..11..201I, ISSN 0033-5606, MR 0003649, doi:10.1093/qmath/os-11.1.201
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[1] 参见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].

[29] 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）.

[30] 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）.

[31] 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）.

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查论编数论中的L函数
解析例子	黎曼ζ函数狄利克雷L函数赫克特征的L函数（英语：Hecke character）自守L函数（英语：Automorphic L-function）塞尔伯格类（英语：Selberg class）
代数例子	戴德金ζ函数（英语：Dedekind zeta function）阿廷L函数（英语：Artin L-function）哈斯-韦伊L函数（英语：Hasse–Weil zeta function）动形L函数（英语：Motivic L-function）
定理	类数公式黎曼-冯·曼戈尔特公式（英语：Riemann–von Mangoldt formula）韦伊猜想（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马努金-皮特森猜想（英语：Ramanujan–Petersson conjecture）阿廷L函数猜想（英语：Artin conjecture (L-functions)）
代数猜想	贝赫和斯维讷通-戴尔猜想 L函数的特殊值（英语：Special values of L-functions）朗兰兹猜想
p进L函数（英语：p-adic L-function）	岩泽理论的主猜想（英语：Main conjecture of Iwasawa theory）塞尔玛群（英语：Selmer group）欧拉系统（英语：Euler system）