In mathematics, the Euler function is given by \[ \phi(q)=\prod_{k=1}^\infty (1q^k). \] Named after Leonhard Euler, it is a prototypical example of a qseries, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis. 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 \[ \phi(q)=\sum_{n=\infty}^\infty (1)^n q^{(3n^2n)/2}. . \] Note that \[ (3n^2n)/2 . \] 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 QPochhammer 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: \[ \ln(\phi(q))=\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{1q^n} . \] 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 \[ b_m=\sum_{nm}\frac{1}{n}= [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] . \] (see OEIS A000203) References Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New YorkHeidelberg: SpringerVerlag, ISBN 9780387901633, MR0434929 Retrieved from "http://en.wikipedia.org/"

