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{{Short description|Mathematical function}}
[[File:Euler function.png|thumb|right|[[Domain coloring]] plot of ϕ on the [[complex plane]]]]
{{other uses|List of topics named after Leonhard Euler}}{{Distinguish|Euler's totient function}}{{No footnotes|date=July 2018}}

In [[mathematics]], the '''Euler function''' is given by
:<math>\phi(q)=\prod_{k=1}^\infty (1-q^k),\quad |q|<1.</math>
Named after [[Leonhard Euler]], it is a model example of a [[q-series|''q''-series]] and provides the prototypical example of a relation between [[combinatorics]] and [[complex analysis]].

==Properties==
The [[coefficient]] <math>p(k)</math> in the [[formal power series]] expansion for <math>1/\phi(q)</math> gives the number of [[Partition of an integer|partitions]] of ''k''.  That is,
:<math>\frac{1}{\phi(q)}=\sum_{k=0}^\infty p(k) q^k</math>
where <math>p</math> is the [[Partition function (number theory)|partition function]].

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

<math>(3n^2-n)/2</math> is a [[pentagonal number]].

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

The Euler function may be expressed as a [[q-Pochhammer symbol|''q''-Pochhammer symbol]]:

:<math>\phi(q) = (q;q)_{\infty}.</math>

The [[logarithm]] of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about ''q''&thinsp;=&thinsp;0, yielding

:<math>\ln(\phi(q)) = -\sum_{n=1}^\infty\frac{1}{n}\,\frac{q^n}{1-q^n},</math>

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

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

where <math>b_n=-\sum_{d|n}\frac{1}{d}=</math> -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see [[OEIS]] [http://oeis.org/A000203/table A000203])

On account of the identity <math>\sigma(n) = \sum_{d|n} d = \sum_{d|n} \frac{n}{d} </math> , where <math>\sigma(n) </math> is the [[Divisor function|sum-of-divisors function]], this may also be written as

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

Also if <math>a,b\in\mathbb{R}^+</math> and <math>ab=\pi ^2</math>, then<ref>Berndt, B. et al. "The Rogers–Ramanujan Continued Fraction"</ref>

:<math>a^{1/4}e^{-a/12}\phi (e^{-2a})=b^{1/4}e^{-b/12}\phi (e^{-2b}).</math>

==Special values==

The next identities come from [[Srinivasa Ramanujan|Ramanujan]]'s Notebooks:<ref>{{Cite book |last1=Berndt |first1=Bruce C. |title=Ramanujan's Notebooks Part V |publisher=Springer |year=1998 |isbn=978-1-4612-7221-2}} p. 326</ref>

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

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

: <math>\phi(e^{-4\pi})=\frac{e^{\pi/6}\Gamma\left(\frac14\right)}{2^{{11}/8}\pi^{3/4}}</math>

: <math>\phi(e^{-8\pi})=\frac{e^{\pi/3}\Gamma\left(\frac14\right)}{2^{29/16}\pi^{3/4}}(\sqrt{2}-1)^{1/4}</math>

Using the [[Pentagonal number theorem]], exchanging sum and [[integral]], and then invoking complex-analytic methods, one derives<ref>{{Cite OEIS|A258232}}</ref>

: <math>\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}.</math>

==References==
{{reflist}}
* {{Apostol IANT}}

{{Leonhard Euler}}

[[Category:Number theory]]
[[Category:Q-analogs]]
[[Category:Leonhard Euler]]