- #1
binbagsss
- 1,254
- 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.