In mathematics, the Euler function is given by Named after Leonhard Euler, it is a model example of a q-series 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/\phi(q) gives the number of partitions of k. That is, where p is the partition function. The Euler identity, also known as the Pentagonal number theorem, is (3n^2-n)/2 is a pentagonal number. The Euler function is related to the Dedekind eta function as 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 where -[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 identity, where \sigma(n) is the sum-of-divisors function, this may also be written as Also if and ab=\pi ^2, then

Special values

The next identities come from Ramanujan's Notebooks: Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives