Euler function

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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_^\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 Maclaurin series for [1/\phi(q)] gives the number of all partitions of k. That is,

[\frac=\sum_^\infty p(k) q^k]

where [p(k)] is the partition function of k.

[\phi(q)=\sum_^\infty (-1)^n q^ ]

Note that [(3n^2-n)/2] is a pentagon number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as

[\phi(q)= q^ \eta(\tau)]

where [q=e^] is the square of the nome.

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

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9

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