I asked ChatGPT about the relationship between the Riemann Xi function and the Jacobi theta functions. ChatGPT provided the following relation:
$\xi\left(\frac{1}{2} + it\right) = \frac{1}{2} \left( \theta_3^4(0, e^{-\pi t^2}) - \theta_4^4(0, e^{-\pi t^2}) \right)$
In the above equation, $\theta_3$ represents the third Jacobi theta function, while $\theta_4$ represents the fourth Jacobi theta function.
$\theta_3(0,\tau) = 1 + 2 \sum_{n=1}^{\infty} e^{i \pi n^2 \tau} $
$\theta_4(0,\tau) = 1 + 2 \sum_{n=1}^{\infty} (-1)^n e^{i \pi n^2 \tau} $
I'm not familiar with this formula. I'm curious to know whether it's a hallucination on ChatGPT's part or if it's a well-established result.
If it is a well-known and proven result, could someone provide a reference for it? I've searched online and also inquired with ChatGPT, but I couldn't find a reference.
There are two notations for theta functions. They are related by
$$ \theta_j(z\vert\tau)=\theta_j(z,e^{\pi i\tau}),\tag1 $$
so it is more correct to write $\theta_j(z|\tau)$ instead of $\theta_j(z,\tau)$. From the identity
$$ \theta_3^4(0\vert\tau)=\theta_2^4(0\vert\tau)+\theta_4^4(0\vert\tau), $$
we see that the OP's identity is equivalent to
$$ \xi\left(\frac12+it\right)=\frac12\theta_2^4(0,e^{-\pi t^2}).\tag2 $$
From the infinite product formula (with $q=e^{-\pi t^2}$)
$$ \theta_2(0,q)=2q^{1/4}\prod_{n\ge1}(1-q^{2n})(1+q^{2n})^2, $$
it is clear that the right hand side has no zeros when $|q|<1$, which clearly contradicts the fact that $\xi\left(\frac12+it\right)$ has infinitely many real zeros (i.e. $\zeta$ has infinitely many zeros on the critical line).