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| In [[mathematics]], '''Apéry's theorem''' is a result in [[number theory]] that states the [[Apéry's constant]] ζ(3) is [[irrational Number|irrational]]. That is, the number
| | == M＆Mは配らウッドヒスイの手鏡 == |
| :<math>\zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}=\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\ldots = 1.2020569\ldots</math>
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| cannot be written as a fraction ''p''/''q'' with ''p'' and ''q'' being integers.
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| ==History==
| | M＆Mは配らウッドヒスイの手鏡 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン 取扱店]。<br><br>は「私を与える？ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン アウトレット] '秦ゆう心李チャン。<br>huanlingミラーを持つ<br>、すべての可能性で、この第二の場所が手にします [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン スニーカー]。DUANMUジェイドは、このM＆M以来ミラーに直接投入<br><br>があります [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン ブーツ]。<br><br>（紳士教会http://www.junzitang.comを提供しています）<br><br>は 'あなたは？'秦Yuは尋ねた。<br><br>「結婚し参加していきます、私は、秦ゆう、あなたは私は神の女王に達すると思いますか？ 'DUANMUジェイド」は、秦の微笑み<br><br>羽が、実際には、私はあなたが初めて競争する場所は、アウト '羅ゆうナイフ」であることを参照してください、私は知っている......あなたがかもしれ<br>18人、唯一の本当の愛JIANGを<br>し、彼女のためにすべてを与えて喜んで。 '<br><br>「JIANGはおそらく幸せになります。あなたと結婚し、M＆Mのミラーは、私が与えたので、私はまた、あなたが成功することができますことを願っています<br>あなたを<br>。 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン サンダル] '<br><br>DUANMUジェイドはまだM＆Mが利きミラー |
| | | 相关的主题文章： |
| [[Euler]] has proven that if ''n'' is a positive integer then | | <ul> |
| :<math>\frac{1}{1^{2n}}+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\ldots = \frac{p}{q}\pi^{2n}</math> | | |
| for some rational number ''p''/''q''. Specifically, writing the infinite series on the left as ζ(2''n'') he showed
| | <li>[http://globalvage.com/home.php?mod=space&uid=61393 http://globalvage.com/home.php?mod=space&uid=61393]</li> |
| :<math>\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}</math>
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| where the ''B<sub>n</sub>'' are the rational [[Bernoulli numbers]]. Once it was proved that π<sup>''n''</sup> is always irrational this showed that ζ(2''n'') is irrational for all positive integers ''n''.
| | <li>[http://meilisheng.cn/plus/view.php?aid=245403 http://meilisheng.cn/plus/view.php?aid=245403]</li> |
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| No such representation in terms of π is known for the so-called [[zeta constants]] for odd arguments, the values ζ(2''n''+1) for positive integers ''n''. It has been conjectured that the ratios of these quantities
| | <li>[http://www.louyuwang.com/bbs/home.php?mod=space&uid=106344 http://www.louyuwang.com/bbs/home.php?mod=space&uid=106344]</li> |
| :<math>\frac{\zeta(2n+1)}{\pi^{2n+1}},</math> | | |
| are transcendental for every integer ''n'' ≥ 1.<ref>{{cite journal |last=Kohnen | first=Winfried |title=Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms |journal=Proc. Indian Acad. Sci. Math. Sci. |volume=99 | number=3 |year=1989 |pages=231–233 |doi=10.1007/BF02864395}}</ref>
| | </ul> |
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| Because of this, no proof could be found to show that the zeta constants with odd arguments were irrational, even though they were—and still are—all believed to be [[transcendental number|transcendental]]. However, in June 1978, [[Roger Apéry]] gave a talk entitled "Sur l'irrationalité de ζ(3)." During the course of the talk he outlined proofs that ζ(3) and ζ(2) were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π. Due to the wholly unexpected nature of the result and Apéry's blasé and very sketchy approach to the subject many of the mathematicians in the audience dismissed the proof as flawed. However [[Henri Cohen (number theorist)|Henri Cohen]], [[Hendrik Lenstra]], and [[Alfred van der Poorten]] suspected Apéry was onto something and set out to confirm his proof. Two months later they finished verification of Apéry's proof, and on August the 18th Cohen delivered a lecture giving full details of the proof. After the lecture Apéry himself took to the podium to explain the source of some of his ideas.<ref>{{cite journal |author=A. van der Poorten |title=A proof that Euler missed... |journal=[[The Mathematical Intelligencer]] |volume=1 |issue=4 |year=1979 |pages=195–203 |doi=10.1007/BF03028234 |url=http://www.maths.mq.edu.au/~alf/45.pdf}}</ref>
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| ==Apéry's proof==
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| Apéry's original proof<ref>{{cite journal | last1 = Apéry | first1 = R. | year = 1979 | title = Irrationalité de ζ(2) et ζ(3) | url = | journal = Astérisque | volume = 61 | issue = | pages = 11–13 }}</ref><ref>{{Citation | last1=Apéry | first1 = R. | year = 1981 | chapter=Interpolation de fractions continues et irrationalité de certaines constantes | title=Bulletin de la section des sciences du C.T.H.S III | pages=37–53 }}</ref> was based on the well known irrationality criterion from [[Peter Gustav Lejeune Dirichlet]], which states that a number ξ is irrational if there are infinitely many coprime integers ''p'' and ''q'' such that
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| :<math>\left|\xi-\frac{p}{q}\right|<\frac{c}{q^{1+\delta}}</math> | |
| for some fixed ''c'',δ>0.
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| The starting point for Apéry was the series representation of ζ(3) as
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| :<math>\zeta(3) = \frac{5}{2} \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^3\binom{2n}{n}}.</math>
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| Roughly speaking, Apéry then defined a sequence ''c<sub>n,k</sub>'' which converges to ζ(3) about as fast as the above series, specifically
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| :<math>c_{n,k}=\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{(-1)^{m-1}}{2m^{3}\binom{n}{m}\binom{n+m}{m}}.</math>
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| He then defined two more sequences ''a<sub>n</sub>'' and ''b<sub>n</sub>'' that, roughly, have the quotient ''c<sub>n,k</sub>''. These sequences were
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| :<math>a_{n}=\sum_{k=0}^{n}c_{n,k}\binom{n}{k}^{2}\binom{n+k}{k}^{2}</math>
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| and
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| :<math>b_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2}.</math>
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| The sequence ''a<sub>n</sub>''/''b<sub>n</sub>'' converges to ζ(3) fast enough to apply the criterion, but unfortunately ''a<sub>n</sub>'' is not an integer after ''n''=2. Nevertheless, Apéry showed that even after multiplying ''a<sub>n</sub>'' and ''b<sub>n</sub>'' by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality.
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| ==Later proofs==
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| Within a year of Apéry's result an alternative proof was found by [[Frits Beukers]],<ref>{{cite journal |author=F. Beukers |title=A note on the irrationality of ζ(2) and ζ(3) |journal=Bulletin of the London Mathematical Society |volume=11 |year=1979 |issue=3 |pages=268–272 |doi=10.1112/blms/11.3.268}}</ref> who replaced Apéry's series with integrals involving the [[shifted Legendre polynomials]] <math>\tilde{P_{n}}(x)</math>. Using a representation that would later be generalized to [[Hadjicostas's formula]], Beukers showed that
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| :<math>\int_{0}^{1}\int_{0}^{1}\frac{-\log(xy)}{1-xy}\tilde{P_{n}}(x)\tilde{P_{n}}(y)dxdy=\frac{A_{n}+B_{n}\zeta(3)}{\operatorname{lcm}\left[1,\ldots,n\right]^{3}}</math>
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| for some integers ''A<sub>n</sub>'' and ''B<sub>n</sub>'' (sequences {{OEIS2C|A171484}} and {{OEIS2C|A171485}}). Using partial integration and the assumption that ζ(3) was rational and equal to ''a''/''b'', Beukers eventually derived the inequality
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| :<math>0<\frac{1}{b}\leq\left|A_{n}+B_{n}\zeta(3)\right|\leq 4\left(\frac{4}{5}\right)^{n},</math>
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| which is a [[Proof by contradiction|contradiction]] since the right-most expression tends to zero and so must eventually fall below 1/''b''.
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| A more recent proof by [[Wadim Zudilin]]<ref>W. Zudilin (2002), [http://arxiv.org/abs/math/0202159 ''An Elementary Proof of Apéry's Theorem''].</ref> is more reminiscent of Apéry's original proof, and also has similarities to a fourth proof by [[Yuri Valentinovich Nesterenko|Yuri Nesterenko]].<ref>{{cite journal |author=Ю. В. Нестеренко |title=Некоторые замечания о ζ(3) |language=Russian |journal=Матем. Заметки |volume=59 |issue=6 |year=1996 |pages=865–880 |url=http://mi.mathnet.ru/mz1785}} English translation: {{cite journal |author=Yu. V. Nesterenko |title=A Few Remarks on ζ(3) |journal=Math. Notes |volume=59 |issue=6 |year=1996 |pages=625–636 |doi=10.1007/BF02307212}}</ref> These later proofs again derive a contradiction from the assumption that ζ(3) is rational by constructing sequences that tend to zero but are bounded below by some positive constant. They are somewhat less transparent than the earlier proofs, relying as they do on hypergeometric series.
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| ==Higher zeta constants==
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| Apéry and Beukers could simplify their proofs to work on ζ(2) as well thanks to the series representation
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| :<math>\zeta(2)=3\sum_{n=1}^{\infty}\frac{1}{n^{2}\binom{2n}{n}}.</math>
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| Due to the success of Apéry's method a search was undertaken for a number ξ<sub>5</sub> with the property that
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| :<math>\zeta(5)=\xi_{5}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{5}\binom{2n}{n}}.</math> | |
| If such a ξ<sub>5</sub> were found then the methods used to prove Apéry's theorem would be expected to work on a proof that ζ(5) is irrational. Unfortunately, extensive computer searching<ref>D. H. Bailey, J. Borwein, N. Calkin, R. Girgensohn, R. Luke, and V. Moll, ''Experimental Mathematics in Action'', 2007.</ref> has failed to find such a constant, and in fact it is now known that if ξ<sub>5</sub> exists and if it is an algebraic number of degree at most 25, then the coefficients in its [[Minimal polynomial (field theory)|minimal polynomial]] must be enormous, at least 10<sup>383</sup>, so extending Apéry's proof to work on the higher odd zeta constants doesn't seem likely to work.
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| Despite this, many mathematicians working in this area expect a breakthrough sometime soon.<ref>{{cite book |author=Jorn Steuding |title=Diophantine Analysis (Discrete Mathematics and Its Applications) |publisher=Chapman & Hall/CRC |location=Boca Raton |year=2005 |pages=280|isbn=978-1-58488-482-8 |oclc= |doi= |accessdate=}}</ref> Indeed, recent work by [[Wadim Zudilin]] and Tanguy Rivoal has shown that infinitely many of the numbers ζ(2''n''+1) must be irrational,<ref>{{cite journal | last1 = Rivoal | first1 = T. | year = 2000 | title = La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs | url = | journal = Comptes Rendus de l'Académie des Sciences. Série I. Mathématique | volume = 331 | issue = | pages = 267–270 |doi = 10.1016/S0764-4442(00)01624-4 |arxiv=math/0008051}}</ref> and even that at least one of the numbers ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.<ref>{{cite journal |author=W. Zudilin |title=One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational |journal=Russ. Math. Surv. |year=2001 |volume=56 |issue=4 |pages=774–776 |doi=10.1070/RM2001v056n04ABEH000427}}</ref> Their work uses linear forms in values of the zeta function and estimates upon them to bound the dimension of a [[vector space]] spanned by values of the zeta function at odd integers. Hopes that Zudilin could cut his list further to just one number did not materialise, but work on this problem is still an active area of research. Higher zeta constants have application to physics: they describe correlation functions in [[Heisenberg model (quantum)|quantum spin chains]]. See for example reference.<ref>{{cite journal |author=H. E. Boos, V. E. Korepin, Y. Nishiyama, M. Shiroishi |title=Quantum Correlations and Number Theory|journal=
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| Journal reference: Journal of Physics A|year=2002 |volume=35 |pages=:4443–4452 }}</ref>
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| ==References==
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| {{reflist|2}}
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| {{DEFAULTSORT:Apery's Theorem}}
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| [[Category:Zeta and L-functions]]
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| [[Category:Theorems in number theory]]
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