[Euler function](https://en.wikipedia.org/wiki/Euler_function) is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod\_{i=1}^\infty(1-x^i)$$


Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
$$\phi(x)<\prod\_{i=1}^2(1-x^i)=1-x-x^2+x^3$$



> 
> **What lower bounds are known for $\phi(x)$**?
> 
> 
> 




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I'm especially interested in bounds which can be computed efficiently as this is needed for a computer application.



Lower bound for Euler's function
[mpmath](http://docs.sympy.org/dev/modules/mpmath/functions/qfunctions.html) can compute it very efficiently.


The function is called `mpmath.qp`. Here is a sage session for $\phi(e^{-\pi})$
with precision 100 decimal digits which takes about 360 microseconds on and old machine:



```
sage: time mpmath.qp(mpmath.exp(-mpmath.pi))
CPU times: user 0 ns, sys: 0 ns, total: 0 ns
Wall time: 361 µs
0.9549187899876741037512339781102910776327153738078052831487991916760940356867145395349815186744610988

```

This agrees with wikipedia's closed form for 
$$\phi(e^{-\pi})=\frac{e^{\pi/24}\Gamma\left(\frac14\right)}{2^{7/8}\pi^{3/4}}$$


