Euler function

In mathematics, the Euler function is given by

${\displaystyle \phi (q)=\prod _{k=1}^{\infty }(1-q^{k}),\quad |q|<1.}$

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 ${\displaystyle p(k)}$ in the formal power series expansion for ${\displaystyle 1/\phi (q)}$ gives the number of partitions of k. That is,

${\displaystyle {\frac {1}{\phi (q)}}=\sum _{k=0}^{\infty }p(k)q^{k}}$

where ${\displaystyle p}$ is the partition function.

The Euler identity, also known as the Pentagonal number theorem, is

${\displaystyle \phi (q)=\sum _{n=-\infty }^{\infty }(-1)^{n}q^{(3n^{2}-n)/2}.}$

${\displaystyle (3n^{2}-n)/2}$ is a pentagonal number.

The Euler function is related to the Dedekind eta function as

${\displaystyle \phi (e^{2\pi i\tau })=e^{-\pi i\tau /12}\eta (\tau ).}$

The Euler function may be expressed as a q-Pochhammer symbol:

${\displaystyle \phi (q)=(q;q)_{\infty }.}$

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

${\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {1}{n}}\,{\frac {q^{n}}{1-q^{n}}},}$

which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as

${\displaystyle \ln(\phi (q))=\sum _{n=1}^{\infty }b_{n}q^{n}}$

where ${\displaystyle b_{n}=-\sum _{d|n}{\frac {1}{d}}=}$ -[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 ${\displaystyle \sigma (n)=\sum _{d|n}d=\sum _{d|n}{\frac {n}{d}}}$ , where ${\displaystyle \sigma (n)}$ is the sum-of-divisors function, this may also be written as

${\displaystyle \ln(\phi (q))=-\sum _{n=1}^{\infty }{\frac {\sigma (n)}{n}}\ q^{n}}$.

Also if ${\displaystyle a,b\in \mathbb {R} ^{+}}$ and ${\displaystyle ab=\pi ^{2}}$, then^[1]

${\displaystyle a^{1/4}e^{-a/12}\phi (e^{-2a})=b^{1/4}e^{-b/12}\phi (e^{-2b}).}$

Special values

The next identities come from Ramanujan's Notebooks:^[2]

${\displaystyle \phi (e^{-\pi })={\frac {e^{\pi /24}\Gamma \left({\frac {1}{4}}\right)}{2^{7/8}\pi ^{3/4}}}}$

${\displaystyle \phi (e^{-2\pi })={\frac {e^{\pi /12}\Gamma \left({\frac {1}{4}}\right)}{2\pi ^{3/4}}}}$

${\displaystyle \phi (e^{-4\pi })={\frac {e^{\pi /6}\Gamma \left({\frac {1}{4}}\right)}{2^{{11}/8}\pi ^{3/4}}}}$

${\displaystyle \phi (e^{-8\pi })={\frac {e^{\pi /3}\Gamma \left({\frac {1}{4}}\right)}{2^{29/16}\pi ^{3/4}}}({\sqrt {2}}-1)^{1/4}}$

Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives^[3]

${\displaystyle \int _{0}^{1}\phi (q)\,\mathrm {d} q={\frac {8{\sqrt {\frac {3}{23}}}\pi \sinh \left({\frac {{\sqrt {23}}\pi }{6}}\right)}{2\cosh \left({\frac {{\sqrt {23}}\pi }{3}}\right)-1}}.}$

References

• 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