Editing Poisson summation formula (section)

===Number theory===
In [[number theory]], Poisson summation can also be used to derive a variety of functional equations including the functional equation for the [[Riemann zeta function]].<ref>[[Harold Edwards (mathematician)|H. M. Edwards]] (1974). ''Riemann's Zeta Function''. Academic Press, pp. 209–11. {{ISBN|0-486-41740-9}}.</ref>

One important such use of Poisson summation concerns [[theta function]]s: periodic summations of Gaussians. Put <math> q= e^{i\pi \tau } </math>, for <math> \tau</math> a complex number in the upper half plane, and define the theta function:

<math display="block"> \theta ( \tau) =  \sum_n q^{n^2}. </math>

The relation between <math> \theta (-1/\tau)</math>  and <math> \theta (\tau)</math>  turns out to be important for number theory, since this kind of relation is one of the defining properties of a [[modular form]].  By choosing <math>s(x)= e^{-\pi x^2}</math> and using the fact that <math>S(f) = e^{-\pi f ^2},</math> one can conclude:

<math display="block">\theta \left({-1\over\tau}\right) = \sqrt{\tau \over i} \theta (\tau),</math> by putting <math>{1/\lambda} = \sqrt{\tau/i}.</math>

It follows from this that <math>\theta^8</math> has a simple transformation property under <math>\tau \mapsto {-1/ \tau}</math> and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of eight perfect squares.