Namespaces
Variants
Actions

Difference between revisions of "Apéry numbers"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (latex details)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
The Apéry numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302602.png" /> are defined by the finite sums
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302603.png" /></td> </tr></table>
+
Out of 29 formulas, 29 were replaced by TEX code.-->
  
for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302604.png" />. They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302606.png" />, respectively. In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302607.png" />, Apéry showed that there exists a sequence of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302608.png" /> with denominator dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026011.png" />. Together with the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026012.png" />, this implies the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026014.png" />. For a very lively and amusing account of Apéry's discovery, see [[#References|[a4]]]. In 1979 F. Beukers [[#References|[a1]]] gave a very short irrationality proof of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026015.png" />, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026016.png" /> so far (2001).
+
{{TEX|semi-auto}}{{TEX|done}}
 +
The Apéry numbers $a _ { n }$, $b _ { n }$ are defined by the finite sums
  
T. Rival [[#References|[a5]]] proved the very surprising result that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026017.png" /> for infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026018.png" />.
+
\begin{equation*} a _ { n } = \sum _ { k = 0 } ^ { n } \left( \begin{array} { c } { n + k } \\ { k } \end{array} \right) ^ { 2 } \left( \begin{array} { c } { n } \\ { k } \end{array} \right) ^ { 2 } , \quad b _ { n } = \sum _ { k = 0 } ^ { n } \left( \begin{array} { c } { n + k } \\ { k } \end{array} \right) \left( \begin{array} { c } { n } \\ { k } \end{array} \right) ^ { 2 } \end{equation*}
  
It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026019.png" /> for all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026021.png" /> and all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026022.png" />. Another congruence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026023.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026024.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026025.png" /> denotes the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026026.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026027.png" />-expansion of a modular cusp form. For more details see [[#References|[a2]]], [[#References|[a3]]].
+
for every integer $n \geq 0$. They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of $\zeta ( 3 )$ and $\zeta ( 2 ) = \pi ^ { 2 } / 6$, respectively. In the case of $\zeta ( 3 )$, Apéry showed that there exists a sequence of rational numbers $c _ { n }$ with denominator dividing $\operatorname { lcm } ( 1 , \ldots , n ) ^ { 3 }$ such that $0 < | a _ { n } \zeta ( 3 ) - c _ { n } | < ( \sqrt { 2 } - 1 ) ^ { 4 n }$ for all $n > 0$. Together with the fact that $\operatorname { lcm } ( 1 , \dots , n ) > 3 ^ { n }$, this implies the irrationality of $\zeta ( 3 )$. For a very lively and amusing account of Apéry's discovery, see [[#References|[a4]]]. In 1979 F. Beukers [[#References|[a1]]] gave a very short irrationality proof of $\zeta ( 3 )$, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of $\zeta ( 5 )$ so far (2001).
 +
 
 +
T. Rival [[#References|[a5]]] proved the very surprising result that $\zeta ( 2 n + 1 ) \notin \mathbf{Q}$ for infinitely many $n$.
 +
 
 +
It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, $a _ { m p ^ r} \equiv a _ { m p ^ { r - 1 } } ( \operatorname { mod } p ^ { 3 r } )$ for all positive integers $m$, $r$ and all prime numbers $p \geq 5$. Another congruence is $a _ {( p - 1 )/ 2 } \equiv \gamma _ { p } ( \operatorname { mod } p )$ for all prime numbers $p \geq 5$. Here, $\gamma _ { n }$ denotes the coefficient of $q ^ { n }$ in the $q$-expansion of a modular cusp form. For more details see [[#References|[a2]]], [[#References|[a3]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Beukers,   "A note on the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026028.png" />"  ''Bull. London Math. Soc.'' , '''11'''  (1979)  pp. 268–272</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Beukers,   "Some congruences for the Apéry numbers"  ''J. Number Theory'' , '''21'''  (1985)  pp. 141–155</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Beukers,   "Another conguence for the Apéry numbers"  ''J. Number Theory'' , '''25'''  (1987)  pp. 201–210</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.J. van der Poorten,   "A proof that Euler missed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026029.png" /> Apéry's proof of the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026030.png" />"  ''Math. Intelligencer'' , '''1'''  (1979)  pp. 195–203</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Rival,   "La fonction zêta de Riemann pren une infinité de valeurs irrationnelles aux entiers impairs"  ''C.R. Acad. Sci. Paris'' , '''331'''  (2000)  pp. 267–270</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  F. Beukers, "A note on the irrationality of $\zeta ( 3 )$"  ''Bull. London Math. Soc.'' , '''11'''  (1979)  pp. 268–272</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  F. Beukers, "Some congruences for the Apéry numbers"  ''J. Number Theory'' , '''21'''  (1985)  pp. 141–155</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top">  F. Beukers, "Another conguence for the Apéry numbers"  ''J. Number Theory'' , '''25'''  (1987)  pp. 201–210</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top">  A.J. van der Poorten, "A proof that Euler missed $...$ Apéry's proof of the irrationality of $\zeta ( 3 )$"  ''Math. Intelligencer'' , '''1'''  (1979)  pp. 195–203</td></tr>
 +
<tr><td valign="top">[a5]</td> <td valign="top">  T. Rivoal, "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs"  ''C.R. Acad. Sci. Paris'' , '''331'''  (2000)  pp. 267–270</td></tr>
 +
</table>

Latest revision as of 20:51, 23 January 2024

The Apéry numbers $a _ { n }$, $b _ { n }$ are defined by the finite sums

\begin{equation*} a _ { n } = \sum _ { k = 0 } ^ { n } \left( \begin{array} { c } { n + k } \\ { k } \end{array} \right) ^ { 2 } \left( \begin{array} { c } { n } \\ { k } \end{array} \right) ^ { 2 } , \quad b _ { n } = \sum _ { k = 0 } ^ { n } \left( \begin{array} { c } { n + k } \\ { k } \end{array} \right) \left( \begin{array} { c } { n } \\ { k } \end{array} \right) ^ { 2 } \end{equation*}

for every integer $n \geq 0$. They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of $\zeta ( 3 )$ and $\zeta ( 2 ) = \pi ^ { 2 } / 6$, respectively. In the case of $\zeta ( 3 )$, Apéry showed that there exists a sequence of rational numbers $c _ { n }$ with denominator dividing $\operatorname { lcm } ( 1 , \ldots , n ) ^ { 3 }$ such that $0 < | a _ { n } \zeta ( 3 ) - c _ { n } | < ( \sqrt { 2 } - 1 ) ^ { 4 n }$ for all $n > 0$. Together with the fact that $\operatorname { lcm } ( 1 , \dots , n ) > 3 ^ { n }$, this implies the irrationality of $\zeta ( 3 )$. For a very lively and amusing account of Apéry's discovery, see [a4]. In 1979 F. Beukers [a1] gave a very short irrationality proof of $\zeta ( 3 )$, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of $\zeta ( 5 )$ so far (2001).

T. Rival [a5] proved the very surprising result that $\zeta ( 2 n + 1 ) \notin \mathbf{Q}$ for infinitely many $n$.

It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, $a _ { m p ^ r} \equiv a _ { m p ^ { r - 1 } } ( \operatorname { mod } p ^ { 3 r } )$ for all positive integers $m$, $r$ and all prime numbers $p \geq 5$. Another congruence is $a _ {( p - 1 )/ 2 } \equiv \gamma _ { p } ( \operatorname { mod } p )$ for all prime numbers $p \geq 5$. Here, $\gamma _ { n }$ denotes the coefficient of $q ^ { n }$ in the $q$-expansion of a modular cusp form. For more details see [a2], [a3].

References

[a1] F. Beukers, "A note on the irrationality of $\zeta ( 3 )$" Bull. London Math. Soc. , 11 (1979) pp. 268–272
[a2] F. Beukers, "Some congruences for the Apéry numbers" J. Number Theory , 21 (1985) pp. 141–155
[a3] F. Beukers, "Another conguence for the Apéry numbers" J. Number Theory , 25 (1987) pp. 201–210
[a4] A.J. van der Poorten, "A proof that Euler missed $...$ Apéry's proof of the irrationality of $\zeta ( 3 )$" Math. Intelligencer , 1 (1979) pp. 195–203
[a5] T. Rivoal, "La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs" C.R. Acad. Sci. Paris , 331 (2000) pp. 267–270
How to Cite This Entry:
Apéry numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ap%C3%A9ry_numbers&oldid=13350
This article was adapted from an original article by Frits Beukers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article