Editing Riemann zeta function (section)

===Theta functions===
The Riemann zeta function can be given by a Mellin transform<ref>{{Cite book |first=Jürgen |last=Neukirch |title=Algebraic number theory |publisher=Springer |date=1999 |page=422 |isbn=3-540-65399-6}}</ref>

:<math>2\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \int_0^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t,</math>

in terms of [[Theta function|Jacobi's theta function]]

:<math>\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2\tau}.</math>

However, this integral only converges if the real part of {{mvar|s}} is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all {{mvar|s}} except 0 and 1:

:<math> \pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}\right)\zeta(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2} \int_0^1 \left(\theta(it)-t^{-\frac12}\right)t^{\frac{s}{2}-1}\,\mathrm{d}t + \frac{1}{2}\int_1^\infty \bigl(\theta(it)-1\bigr)t^{\frac{s}{2}-1}\,\mathrm{d}t.</math>