Cantor's product

In analysis, a Cantorian product is an infinite product whose terms consist of rational numbers of the form , whereby the denominators appearing in them are always natural numbers and, moreover, always such that the denominator of the -th term is always at least as large as that Square of the denominator belonging to the preceding term ${\displaystyle 1+{\tfrac {1}{q}}}$ ${\displaystyle q_{n+1}}$ ${\displaystyle n+1}$ ${\displaystyle (n\in \mathbb {N} )}$ ${\displaystyle n}$ ${\displaystyle q_{n}}$

The Cantor products were introduced by Georg Cantor in a paper from 1869. As Cantor showed in it, any real number can be represented in the form of a Cantor product. Euler's product equation, which goes back to Leonhard Euler , is fundamental for Cantor's explanations ${\displaystyle {\alpha }_{0}>1}$

${\displaystyle {\frac {1}{1-x}}=\prod _{n=0}^{\infty }{(1+x^{2^{n}})}}$,

which is valid for all real (and moreover even for all complex ) numbers of the absolute value. ${\displaystyle x}$ ${\displaystyle |x|<1}$

Cantor's sentence

Cantor's sentence about Cantor's products can be summarized as follows:

Be a real number. Then: ${\displaystyle {\alpha }_{0}>1}$

(I) To can be one and only one sequence of natural numbers so determined that a product representation of the form ${\displaystyle {\alpha }_{0}}$ ${\displaystyle (q_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle {\alpha }_{0}}$

${\displaystyle {\alpha }_{0}=\prod _{n=0}^{\infty }\left(1+{\tfrac {1}{q_{n}}}\right)}$

has, whereby in this number sequence the inequality is fulfilled for every index and there are also only finitely many sequence elements . ${\displaystyle n}$ ${\displaystyle q_{n+1}\geq {q_{n}}^{2}}$ ${\displaystyle q_{n}=1}$

(II) Every Cantor product, i.e. every infinite product of the form described in (I), is convergent .

(III) is a rational number if and only if in Cantor's product representation according to (I) there is always an identity for all subsequent indices from one index onwards . ${\displaystyle {\alpha }_{0}}$ ${\displaystyle N}$ ${\displaystyle n\geq N}$ ${\displaystyle q_{n+1}={q_{n}}^{2}}$

Algorithm for determining the Cantorian product representation

The sequence of numbers can be determined inductively based on as follows : ${\displaystyle (q_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle {\alpha }_{0}>1}$

${\displaystyle q_{0}=\left\lfloor {\frac {{\alpha }_{0}}{{\alpha }_{0}-1}}\right\rfloor }$ and for ${\displaystyle {\alpha }_{n}={\frac {{\alpha }_{n-1}}{1+{\frac {1}{q_{n-1}}}}}\;\;\;q_{n}=\left\lfloor {\frac {{\alpha }_{n}}{{\alpha }_{n}-1}}\right\rfloor }$ ${\displaystyle n\in \mathbb {N} }$

Examples

• Formula from Engel :

For true always ${\displaystyle q\in \mathbb {N} }$

${\displaystyle {\sqrt {\frac {q+1}{q-1}}}=\prod _{n=0}^{\infty }{(1+{\frac {1}{q_{n}}})}}$

with and . ${\displaystyle q_{0}=q\in \mathbb {N} }$ ${\displaystyle q_{n+1}=2\cdot {{q_{n}}^{2}}-1}$ ${\displaystyle (n=0,1,2,3\ldots )}$

The following applies in particular : ${\displaystyle q=3}$

${\displaystyle {\sqrt {2}}={(1+{\frac {1}{3}})}\cdot {(1+{\frac {1}{17}})}\cdot {(1+{\frac {1}{577}})}\cdot {(1+{\frac {1}{665857}})}\cdot \cdots }$

• More examples from Cantor:

${\displaystyle {\sqrt {3}}={(1+{\frac {1}{2}})}\cdot {(1+{\frac {1}{7}})}\cdot {(1+{\frac {1}{97}})}\cdot {(1+{\frac {1}{18817}})}\cdot \cdots }$

${\displaystyle {\sqrt {5}}={(1+{\frac {1}{1}})}\cdot {(1+{\frac {1}{9}})}\cdot {(1+{\frac {1}{161}})}\cdot {(1+{\frac {1}{51841}})}\cdot \cdots }$

${\displaystyle {\sqrt {15}}={(1+{\frac {1}{1}})}\cdot {(1+{\frac {1}{2}})}\cdot {(1+{\frac {1}{4}})}\cdot {(1+{\frac {1}{31}})}\cdot {(1+{\frac {1}{1921}})}\cdot \cdots }$

annotation

• In the first volume of the Lexicon of Mathematics , finite products, which otherwise meet the two secondary conditions mentioned above, are treated as Cantor's products. In addition, it is required for everyone . ${\displaystyle n}$ ${\displaystyle q_{n}\geq 2}$
• Perron mentions that Cantor's products in the irrational numbers converge very quickly. From them one can therefore obtain very good approximate fractions for all real numbers> 1 with just a few calculation steps .
• Two other notable Euler's product representations go back to Euler , namely the following two, which are derived in modern function theory by means of theta functions :

For each complex number of the amount applies: ${\displaystyle z}$ ${\displaystyle |z|<1}$

${\displaystyle \prod _{n=1}^{\infty }{(1-z^{n})}=\sum _{n={-\infty }}^{+\infty }{(-1)^{n}\cdot z^{\frac {3n^{2}+n}{2}}}}$

such as

${\displaystyle \prod _{n=1}^{\infty }{(1-z^{n})^{3}}=\sum _{n={1}}^{\infty }{(-1)^{n-1}\cdot (2n-1)\cdot z^{\frac {n^{2}-n}{2}}}}$ .

literature

• Jonathan M. Borwein , Peter B. Borwein : Pi and the AGM . A Study in Analytic Number Theory and Computational Complexity (=  Canadian Mathematical Society series of monographs and advanced texts . Volume 4 ). John Wiley & Sons, New York 1987, ISBN 0-471-83138-7 .
• Georg Cantor: Two sentences about a certain decomposition of numbers into infinite products . In: Journal of Mathematics and Physics . tape 14 , 1869, pp. 152–158 ( gdz.sub.uni-goettingen.de ).
• Georg Cantor: Collected treatises of mathematical and philosophical content . Reprint of the Berlin 1932 edition. Springer Verlag, Berlin / New York 1980, ISBN 3-540-09849-6 ( MR0616083 ).
• Oskar Perron : Irrational Numbers (=  Göschen's teaching library: Group 1, Pure and Applied Mathematics . Volume 1 ). 4th revised and supplemented edition. Walter de Gruyter Verlag, Berlin 1960 ( MR0115985 ).
• Adolf Hurwitz : Lectures on general function theory and elliptic function . Edited and supplemented by a section on geometric function theory by R. Courant . With an appendix by H. Röhrl (=  The Basic Teachings of Mathematical Sciences in Individual Representations . Volume 3 ). 4th, increased and improved edition. Springer Verlag, Berlin (among others) 1964.
• Guido Walz [Red.]: Lexicon of Mathematics in six volumes . tape 1 . Spectrum Academic Publishing House, Heidelberg / Berlin 2002, ISBN 3-8274-0303-0 .

References and comments

1. ↑ ^{a b c d e} Oskar Perron : Irrational Numbers (=  Göschen's teaching library: Group 1, Pure and Applied Mathematics . Volume 1 ). 4th revised and supplemented edition. Walter de Gruyter Verlag, Berlin 1960, p. 128 ff . ( MR0115985 ).
2. ^ Lexicon of Mathematics in six volumes . tape 1 , p. 278 .
3. ↑ ^{a b c} Cantor: Collected treatises ... p. 43 ff .
4. ↑ is the Gaussian bracket function . ${\displaystyle x\mapsto \lfloor x\rfloor }$
5. ↑ This product representation by also appears in Cantor's work. Cantor made a calculation error and incorrectly indicated the correct value instead . Perron states the correct value for this in the irrational numbers . ${\displaystyle {\sqrt {2}}}$ ${\displaystyle q_{3}=665857=2*577^{2}-1}$ ${\displaystyle q_{3}=667967}$
6. ↑ Even with Cantor had made a miscalculation, because he called instead of the correct value incorrectly . ${\displaystyle {\sqrt {3}}}$ ${\displaystyle q_{3}=18817}$ ${\displaystyle q_{3}=17617}$
7. ^ Hurwitz-Courant: Theory of functions (§ 11) . S. 207 .
8. ↑ Borwein-Borwein: Pi ... (Ch. 3.1) . S. 64-65 .
9. ↑ According to Borwein-Borwein, this is Euler's pentagonal number theorem .