Euler function

In mathematics, the Euler function is given by

Modulus of ϕ on the complex plane, colored so that black = 0, red = 4

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.

PropertiesEdit

The coefficient   in the formal power series expansion for   gives the number of partitions of k. That is,

 

where   is the partition function.

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

 

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   this may also be written as

 

Special valuesEdit

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

 

ReferencesEdit

  • Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001