Modulus of phi on the complex plane, colored so that black=0, red=4
In mathematics, the Euler function is given by
Named after Leonhard Euler, it is a model example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis.
The coefficient in the formal power series expansion for gives the number of all partitions of k. That is,
where is the partition function of k.
The Euler identity, also known as the Pentagonal number theorem is
Note that is a pentagonal 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.
The Euler function may be expressed as a Q-Pochhammer symbol:
The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q=0, yielding:
which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as:
- -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see OEIS A000203)
On account of the following identity,
this may also be written as
The next identities come from Ramanujan's lost notebook, Part V, p. 326.
Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives