Euler function

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title: "Euler function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["number-theory", "q-analogs", "leonhard-euler"] description: "Mathematical function" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Euler_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical function ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f1/Euler_function.png" caption="[[Domain coloring]] plot of ϕ on the [[complex plane"] ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/08/Euler_function_phi(x).svg" caption="Euler function \phi(x)."] ::

In mathematics, the Euler function is given by :\phi(q)=\prod_{k=1}^\infty (1-q^k),\quad |q| 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, :\frac{1}{\phi(q)}=\sum_{k=0}^\infty p(k) q^k where p is the partition function.

The Euler identity, also known as the Pentagonal number theorem, is :\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}.

(3n^2-n)/2 is a pentagonal number.

The Euler function is related to the Dedekind eta function as :\phi (e^{2\pi i\tau})= e^{-\pi i\tau/12} \eta(\tau).

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

:\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

:\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

:\ln(\phi(q)) = \sum_{n=1}^\infty b_n q^n

where 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 \sigma(n) = \sum_{d|n} d = \sum_{d|n} \frac{n}{d} , where \sigma(n) is the sum-of-divisors function, this may also be written as

:\ln(\phi(q)) = -\sum_{n=1}^\infty \frac{\sigma(n)}{n}\ q^n.

Also if a,b\in\mathbb{R}^+ and ab=\pi ^2, then

: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:

: \phi(e^{-\pi})=\frac{e^{\pi/24}\Gamma\left(\frac14\right)}{2^{7/8}\pi^{3/4}}

: \phi(e^{-2\pi})=\frac{e^{\pi/12}\Gamma\left(\frac14\right)}{2\pi^{3/4}}

: \phi(e^{-4\pi})=\frac{e^{\pi/6}\Gamma\left(\frac14\right)}{2^

: \phi(e^{-8\pi})=\frac{e^{\pi/3}\Gamma\left(\frac14\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

: \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

References

1. Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"
2. (1998). "Ramanujan's Notebooks Part V". Springer.
3. {{Cite OEIS. A258232

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