- #1

binbagsss

- 1,266

- 11

## Homework Statement

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.

## Homework Equations

the above

## The Attempt at a Solution

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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.