Euler function: Information from Answers.com

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

For other meanings, see List of topics named after Leonhard Euler.

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 $1 / φ(q)$ 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.

$\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.$

Note that $(3 n 2 - n) / 2$ 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 $q = e 2π i τ$ is the square of the nome.

Note that both functions have the symmetry of the modular group.

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, MR 0434929, ISBN 978-0-387-90163-3

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