黎曼猜想，首先由數學家波恩哈德·黎曼（1826--1866）在1859年提出，是數學中其中一個最著名和最重要而又未解決的問題。一個世紀以來它仍未被解答，即使吸引著很多出色數學家為它苦惱。對比業餘數學家，它對專業數學家更具吸引力。
黎曼猜想(RH)是關於黎曼ζ函數ζ(s)的根分佈的猜想。黎曼ζ函數在任何複數s ≠ 1上也有定義。它在負偶數上也有零（i.e. 當 s = −2, s = −4, s = −6, ...），這些也是「平凡零點」。黎曼猜想關心的，是非平凡零點，及指出：
黎曼ζ函數非平凡零點的實數部份也是½
所以非平凡零點都應該位於直線½ + it上，t為一實數而i為虛數基本單位。沿臨界綫的黎曼ζ函數有時通過Z-函數進行研究, 它的實零點對應于ζ函數在臨界綫上的零點.
素数在自然数中的分布问题在纯粹数学和应用数学上都是很重要的问题。素数在 自然数域中分布并没有一定规则。黎曼（1826--1866）发现素数出现的频率与所 谓黎曼ζ函數紧密相关，
1901年 Helge von Koch 指出，黎曼猜想與敘述
等價。 现在已经验证了最初的1,500,000,000个解，猜想都是正确的。但是否对所有解是 正确的，却没有证明，随着费马最後定理的获证，黎曼猜想作为最困难的数学问题 的地位更加突出。
黎曼猜想是當代數學其中一個最重要而又未解決的問題，主要因為很多深入和重要的結果能在它成立的大前提下被證明。大部份數學家也相信黎曼猜想是正確的（約翰·艾登瑟·利特伍德與塞爾伯格曾提出懷疑。塞爾伯格在晚年時降低了他的懷疑，他在1989年的一篇論文中猜測黎曼猜想對更廣的一類函數也當成立。）克雷數學研究所曾設立了$1,000,000美元的奬金予第一個正確的證明。
歷史黎曼猜想傳統的表達式隠藏了這個猜想的真正重要性。黎曼ζ函數與素數的分佈有著深厚的連結。Helge von Koch在1901年證明了黎曼猜想等價於素數定理一個可觀的強化：給出任何 ε > 0，我們有
式中π(x) 為prime-counting函數，ln(x) 為 x 的自然對數，以及右手邊用上了大O符號
For many global L-functions of function fields (but not number fields), the Riemann hypothesis has been proven. For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), has absolute value
√q
is actually an instance of the Riemann hypothesis in the function field setting.
黎曼猜想與素數
黎曼猜想的實際用途包括一些在黎曼猜想成立前提底下能被證明為真的命題，當中有些更被證明了跟黎曼猜想等價。其中一個就是以上素數定理誤差項的增長率。
黎曼猜想之結果及其等價命題其中一個命題牽涉了默比烏斯函數 μ。命題「等式
在s的實部大於½的時候成立，而且右邊項的和收斂」就等價於黎曼猜想。由此我們能夠總結出假如Mertens 函數的定義為
那聲稱「
對於任何
」就等價於黎曼猜想。這將會對於M的增長給出了一個綳緊的限制，因為即使沒有黎曼猜想我們也能得出
（關於這些符號的意思，見大O符號。）
默比烏斯函數的增長率黎曼猜想等價於一些除μ(n)以外一些積性函數增長率的猜想。例如，因數函數σ(n)由下式給出：
那在n > 5040的時候，
這名為Robin定理並在1984年以Guy Robin命名。另一個有關的上限在2002年由Jeffrey Lagarias提出，他證明了黎曼猜想等價於命題「對於任意自然數 n，
而H
n為第n個harmonic number。
积性函数增长率The Riesz criterion was given by Marcel Riesz in 1916, to the effect that the relation
holds for all ε > 0 if and only if RH holds
for all ε > 0 and some constant C
ε depending on ε. Entering into the proof is the Mobius function μ(n), and so similar results hold for binomial sums over ζ(s − 1) / ζ(s),
and so on, which correspond to Dirichlet series for Euler's totient function, the divisor function, and so on.
Riesz criterion, binomial sums
Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis. Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemann hypothesis.
Weil's criterion, Li's criterion另外兩個跟黎曼猜想等價的命題牽涉了法里數列。假如F
n是法里數列中的第n項，由1/n開始而終於1/1，那命題「給出任何 e > ½
」等價於黎曼猜想。在這裏
是法里數列中n階項的數目。類似地等價於黎曼猜想的命題是「給出任何e > −1.
」
跟法里數列的關係黎曼猜想等價於群論中的一些猜想。舉例說，g(n)，是對稱群S
n 的所有元素的秩之中，最大的一個，也就是蘭道函數, then the Riemann hypothesis is equivalent to the bound, for all n greater than some M, of
跟群論的關係黎曼猜想等價於命題「ζ(s)的導函數ζ'(s)，在區域
上無零點。」
That ζ has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line, so under the usual hypotheses on the Riemann zeta-function we can extend the zero-free region to
. This approach has been fruitful; refining it allowed Norman Levinson to prove his strengthening of the critical line theorem.
临界线定理一些比黎曼猜想強的猜想曾被提出，但它們有被證否的趨勢。Paul Turan證明了假如數列總和
當s大於1時沒有零點，則黎曼猜想成立，但Hugh Montgomery證明了這前提並不成立。另一個更強的猜想，Mertens conjecture，也同樣被證否。
已證否的猜想
相對弱的猜想黎曼猜想有各種比較弱的結果；其中一個是在ζ函數於臨界線上的增長速度上的
Lindelöf猜想，表明了給出任意的e > 0，當t趨向無限，
記第 n 個素數為 p
n ，一個由Albert Ingham得出的結果顯示，Lindelöf猜想將推導出「給出任意 e > 0，
p
n+1 - p
n < p
假如 n 足夠大.」 However, this result is worse than that of the large prime gap conjecture, stated below.
Lindelöf猜想另一個猜想是large prime gap conjecture。哈拉爾德·克拉梅爾證明了，假設黎曼猜想成立，素數 p與其後繼者之間的prime gap將會為
。平均來說，該gap只謹謹為O(lnp) and numerical evidence does not suggest it can grow nearly as fast as the Riemann hypothesis seems to allow, much less as fast as the best that can at present be shown without it.
Large prime gap conjecture
過去數十年很多數學家隊伍聲稱證明了黎曼猜想，而截至2007年為止有少量的證明還沒被驗證。但它們都被數學社群所質疑，而專家們多數並不相信它們是正確的。Matthew R. Watkins from the University of Exeter has a compilation of such claims (serious and ludicrous alike), 而一些其它的證明可在arXiv數據庫中找到。
證明黎曼猜想的嘗試主条目：希爾伯特-波利亞猜想
長久以來，人們猜測黎曼猜想的「正解」是找到一個適當的自伴算符，再由實特徵值的判準導出 ζ(s) 零點實部的資訊。在此方向上已有許多工作，卻仍未有決定性的進展。
黎曼ζ函數的統計學性質與隨機矩陣的特徵值有許多相似處。這為希爾伯特-波利亞猜想提供了一些支持。
在1999年，Michael Berry 與 Jon Keating 猜想經典哈密頓函數 H = xp 有某個未知的量子化
，使得下式成立
更奇特的是，黎曼ζ函數的零點與算子
的譜相同。正則量子化的情形則相反：正則量子化引致海森堡測不準原理 [x,p] = 1 / 2，並使量子諧振子的譜為自然數。重點在於，所求的哈密頓算符應當是個閉自伴算符，方能滿足希爾伯特-波利亞猜想之要求。
跟算子理论的可能關聯
關於計算上找尋ζ函數零點越多越好的嘗試，已經有一段很長的歷史了。其中一個出名的嘗試乃ZetaGrid，一個分散式計算的計劃，一天可檢查上十億個零點。這計劃在2005年11月終止。直至2006年，沒有計算計劃成功找到黎曼猜想的一個反例。
2004年，Xavier Gourdon與Patrick Demichel透過Odlyzko-Schönhage algorithm驗證了黎曼猜想的頭十兆個非平凡零點。
Michael Rubinstein給了公眾一個算法去算出零點。
搜尋ζ函數的零點
參考文獻
Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, (1859) Monatsberichte der Berliner Akademie. (This site provides both a facsimile of the original manuscripts, as well as English translations.)
Jacques Hadamard, Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques, Bulletin Société Mathématique de France 14 (1896) pp 199-220.
歷史文獻
H. M. Edwards, Riemann's Zeta Function, Academic Press, 1974. (Reprinted by Dover Publications, 2001 ISBN 0-486-41740-9)
E. C. Titchmarsh, The Theory of the Riemann Zeta Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
Jeffrey Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, American Mathematical Monthly (2002), no. 109, 534--543（論及與調和數的關聯）
Computation of zeros of the Zeta function (2004). （關於GUE猜想的評論，兼具豐富的書目資料。）
Schoenfeld, Lowell. "Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II." Mathematics of Computation 30 (1976), no. 134, 337--360.
Conrey, J. Brian. "the Riemann Hypothesis" Notices of the American Mathematical Society, March 2003, 341-353. 可自由下載。
現代技術參考
Clay Mathematics Institute, Millennium Problems, (2000) (Announcement of the million dollar rewards for solutions to famous problems in mathematics)
Marcus du Sautoy, The Music of the Primes, HarperCollins, 2003
Marcus du Sautoy, "Prime Numbers Get Hitched", Seed Magazine" (03/27/2006)
Daniel Rockmore, Stalking the Riemann Hypothesis : The Quest to Find the Hidden Law of Prime Numbers, Pantheon Books, New York, 2005. ISBN 0-375-42136-X.
John Derbyshire, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Joseph Henry Press (April 23, 2003), ISBN 0-309-08549-7. 448 page book at a non-specialist level, can be read online for free.
Zetagrid (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed in November 2005.
Ron Howard, A Beautiful Mind (2001). In the film, the mathematician John Nash attempts to prove the Riemann Hypothesis.
Law & Order: Criminal Intent: In season 2, episode 2, "Bright Boy", the detective played by Vincent D'Onofrio tricks a father into confessing to a murder by telling him his son has solved the Riemann Hypothesis.
Numb3rs: In season 1 episode 5, "Prime Suspect", (CBS) Criminals kidnap a mathematician's daughter and demand his allegedly complete proof of Riemann's Hypothesis as ransom.
Ed Pegg, Jr., Ten Trillion Zeta Zeros, (2004) Math Games website. A discussion of Xavier Gourdon's calculation of the first ten trillion non-trivial zeros.
de Vries, The Graph of the Riemann Zeta function ζ(s) (2004). A simple animated java applet.
Erica Klarreich, "Prime Time", New Scientist - November 11, 2000, p. 32. A simple introduction to the Riemann Hypothesis, and its connection to prime numbers and quantum systems.
QEDen A wiki dedicated to solving the millennium problems
Perplex City, an Alternate Reality Game soliciting answers to various puzzles from its players, is offering 60 points to any player to submit a proof of the Riemann Hypothesis on card 238 of Season 1, entitled "Riemann".
Proof, a film starring Gwynneth Paltrow and Anthony Hopkins, in which a character is purported to have solved a problem that sounds like the Riemann Hypothesis.
In the novel PopCo the main character's grandmother used to spend all her free time trying to prove the Riemann Hypothesis.
In Thomas Pynchon's novel Against The Day the Riemann Hypothesis and the question of non-trivial zeros are recurring tropes and preoccupations of the students and spies gathered at Göttingen University in the years preceding World War I.
Karl Sabbagh, The Riemann Hypothesis: the greatest unsolved problem in mathematics, (2003) Farrar, Straus and Giroux, ISBN 0-374-25007-3. Also (2004) First American paperback edition. Conversations with mathematicians working on the problem. The reader should not expect to learn anything about the problem itself.
Dirk L. van Krimpen, Proving the Riemann Hypothesis and other simple things, (2007) The main story of this bundle of math oriented science fiction stories shows the internal geometry of the zeta function in various pictures. From this geometry rules can be deducted proving that indeed for every possible non-trivial zero the real part cannot be anything else than 1/2.
受歡迎的參考資料
Bollobas, Bela, foreword to Littlewood's Miscellany, Cambridge University Press, 1986
Cited References
（中文）Riemann猜想漫談--盧昌海
