You can find a huge collection of examples in this website:

* [Number Theory and Physics Archive](http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm)

As for a concrete example, the critical temperature of the Bose-Einstein condensate is

$$T=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{mk_B}$$

where $\zeta(s)$ is the Riemann zeta function.

Particularly after the work of Ketterle, this seems as "real" and non-speculative as it gets.