Euler function

In mathematics, the Euler function is given by

\[ \phi(q)=\prod_{k=1}^\infty (1-q^k). \]

Named after Leonhard Euler, it is a prototypical 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=1, 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:

\[ \ln(\phi(q))=\sum_{m=1}^\infty b_m q^m . \]

where

\[ b_m=-\sum_{n|m}\frac{1}{n}= -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] . \] (see OEIS A000203)

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, MR0434929