Euler function
n
mathematics, the Euler function is given by
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.
Encyclopedia
- For other meanings, see List of topics named after Leonhard Euler.
In
mathematics, the
Euler function is given by
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 in the
Maclaurin series for gives the number of all
partitions of k. That is,
where is the partition function of k.
The
Euler identity is
Note that is a
pentagon number.
The Euler function is related to the
Dedekind eta function through a
Ramanujan identity as
where is the square of the nome.
Note that both functions have the symmetry of the
modular group.
References
- Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York. ISBN 0-387-90163-9
