# Homework Help: Jacobi Theta Function modularity translation quick q

1. Dec 3, 2016

### binbagsss

1. The problem statement, all variables and given/known data

I have the Jacobi theta series: $\theta^{m}(\tau) = \sum\limits^{\infty}_{n=0} r_{m}(\tau) q^{n}$,

where $q^{n} = e^{2\pi i n \tau}$ and I want to show that $\theta^{m}(\tau + 1) = \theta^{m}(\tau)$

(dont think its needed but) where $r_{m} =$ number of ways of writing $m$ as the sum of $n$ squares.

2. Relevant equations

the above

3. The attempt at a solution

so i get an extra $e^{2\pi i n}$ factor, $\theta^{m}(\tau) = \sum\limits^{\infty}_{n=0}r_{m}(\tau) q^{n} e^{2\pi i n}$

I think it should be obvious what to do now, but I don't know what to do next?

Something like defining a new Fourier coefficient, $r'{m}= r_{m} e^{2\pi i n}$ and then since the sum is to ? but that doesn't seem proper enough?

Many thanks.

2. Dec 3, 2016

### stevendaryl

Staff Emeritus
Hmm. You know that if $n$ is an integer, $e^{i 2n\pi} = 1$?

3. Dec 3, 2016

### binbagsss

oh my ! thank you ha :)