Freeman Dyson's identity for the modular discriminant $\Delta$

286 Views Asked by At

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity :

$$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i \pmod 5 }} \left(\frac{1}{288}\prod_{1\leq i<j\leq 5}(x_i-x_j)\right)q^{(x_1^2+x_2^2+x_3^2+x_4^2+x_5^2)/10}$$ with $q:=e^{2i\pi \tau}$. I would be very interested in a proof of that result.

Thanks everyone !