Euler function

Modulus of phi on the complex plane, colored so that black=0, red=4

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.

Properties

The coefficient $p(k)$ in the formal power series expansion for $1/\phi (q)$ gives the number of all partitions of k. That is,

${\frac {1}{\phi (q)}}=\sum _{{k=0}}^{\infty }p(k)q^{k}$

where $p(k)$ is the partition function of k.

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:

$\phi (q)=(q;q)_{\infty }$

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:

$\ln(\phi (q))=-\sum _{{n=1}}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{1-q^{n}}}$

which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as:

$\ln(\phi (q))=\sum _{{n=1}}^{\infty }b_{n}q^{n}$

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 following identity,

$\sum _{{d|n}}d=\sum _{{d|n}}{\frac nd}$

this may also be written as

$\ln(\phi (q))=-\sum _{{n=1}}^{\infty }{\frac {q^{n}}{n}}\sum _{{d|n}}d$

Special values

The next identities come from Ramanujan's lost notebook, Part V, p. 326.

$\phi (e^{{-\pi }})={\frac {e^{{\pi /24}}\Gamma \left({\frac 14}\right)}{2^{{7/8}}\pi ^{{3/4}}}}$

$\phi (e^{{-2\pi }})={\frac {e^{{\pi /12}}\Gamma \left({\frac 14}\right)}{2\pi ^{{3/4}}}}$

$\phi (e^{{-4\pi }})={\frac {e^{{\pi /6}}\Gamma \left({\frac 14}\right)}{2^{{{11}/8}}\pi ^{{3/4}}}}$

$\phi (e^{{-8\pi }})={\frac {e^{{\pi /3}}\Gamma \left({\frac 14}\right)}{2^{{29/16}}\pi ^{{3/4}}}}({\sqrt {2}}-1)^{{1/4}}$

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