In mathematics, the Euler function is given by
\( \phi (q)=\prod _{{k=1}}^{\infty }(1-q^{k}). \)
Named after Leonhard Euler, it is a model example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis.
Modulus of ϕ on the complex plane, colored so that black = 0, red = 4
Properties
The coefficient \( p(k) \) in the formal power series expansion for \( 1/\phi (q) \) gives the number of partitions of k. That is,
\( {\frac {1}{\phi (q)}}=\sum _{{k=0}}^{\infty }p(k)q^{k} \)
where p {\displaystyle p} p is the partition function.
The Euler identity, also known as the Pentagonal number theorem, is
\( \phi (q)=\sum _{{n=-\infty }}^{\infty }(-1)^{n}q^{{(3n^{2}-n)/2}}. \)
Note that \( (3n^{2}-n)/2 \) is a pentagonal number.
The Euler function is related to the Dedekind eta function through a Ramanujan identity as
\( \phi (q)=q^{{-{\frac {1}{24}}}}\eta (\tau ) \)
where \( q=e^{{2\pi i\tau }} \) is the square of the nome. Note that both functions have the symmetry of the modular group.
The Euler function may be expressed as a q-Pochhammer symbol:
\( {\displaystyle \phi (q)=(q;q)_{\infty }.} \)
The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q = 0, yielding
\( {\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{1-q^{n}}},} \)
which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as
\( {\displaystyle \ln(\phi (q))=\sum _{n=1}^{\infty }b_{n}q^{n}} \) \)
where \( b_{n}=-\sum _{{d|n}}{\frac {1}{d}}= -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] \) (see OEIS A000203)
On account of the identity \( {\displaystyle \sum _{d|n}d=\sum _{d|n}{\frac {n}{d}},} \) this may also be written as
\( {\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {q^{n}}{n}}\sum _{d|n}d.} \)
Special values
The next identities come from Ramanujan's lost notebook, Part V, p. 326.
\( {\displaystyle \phi (e^{-\pi })={\frac {e^{\pi /24}\Gamma \left({\frac {1}{4}}\right)}{2^{7/8}\pi ^{3/4}}}} \)
\( {\displaystyle \phi (e^{-2\pi })={\frac {e^{\pi /12}\Gamma \left({\frac {1}{4}}\right)}{2\pi ^{3/4}}}} \)
\( {\displaystyle \phi (e^{-4\pi })={\frac {e^{\pi /6}\Gamma \left({\frac {1}{4}}\right)}{2^{{11}/8}\pi ^{3/4}}}} \)
\( {\displaystyle \phi (e^{-8\pi })={\frac {e^{\pi /3}\Gamma \left({\frac {1}{4}}\right)}{2^{29/16}\pi ^{3/4}}}({\sqrt {2}}-1)^{1/4}} \)
Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives
\( {\displaystyle \int _{0}^{1}\phi (q){\text{d}}q={\frac {8{\sqrt {\frac {3}{23}}}\pi \sinh \left({\frac {{\sqrt {23}}\pi }{6}}\right)}{2\cosh \left({\frac {{\sqrt {23}}\pi }{3}}\right)-1}}.} \)
References
Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001
Hellenica World - Scientific Library
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