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Let $p$ be an odd prime. Denote $$S_p=\frac{\sqrt p}{2^{(p-1)/2}}\prod_{k=1}^{(p-1)/2}\csc\frac{\pi k^2}p.$$ I conjecture that:
(i) For $p\equiv1\pmod4$, $S_p$ is the fundamental unit of field $\mathbb Q[\sqrt p]$.
(ii) For $p\equiv3\pmod4$, $S_p=\pm1$.
Can we prove or disprove them? Moreover, (iii) can we calculate the sign of $S_p$ when $p\equiv3\pmod4$?

I have made some progress proving (ii).
When $p\equiv3\pmod4$, $$S_p^2=\frac{p}{2^{p-1}}\left(\prod_{k=1}^{(p-1)/2}\csc\frac{\pi k^2}p\right)^2$$ Using some basic facts in number theory, we know $\pm k^2\bmod p$ runs over $1$ to $p-1$ when $k=1,\ldots,(p-1)/2$. Therefore we have $$S_p^2=\frac{p}{2^{p-1}}\prod_{k=1}^{p-1}\csc\frac{\pi k}p=1.$$ But I have no idea how to solve (i) and (iii).

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Your conjecture on $p\equiv 1 \pmod{4}$ is correct except missing the class number, making the smallest counterexample at $p=229$. It follows from the class number formula.


Consider the case $p\equiv 1 \pmod{4}$, $K=\mathbb{Q}(\sqrt{p})$, $\varepsilon > 1$ be the fundamental unit, $K$ is associated with the quadratc character $\chi = (\cdot | p)$. Factorization of Dedekind zeta function $\zeta_K(s) = \zeta(s) L(s,\chi)$ implies

$$ \frac{h \log \varepsilon}{\sqrt{p}} = L(1,\chi) = -\frac{2G(\chi)}{p} \sum_{k=1}^{(p-1)/2} \chi(k)\log(\sin \frac{k \pi}{p})$$ where $h$ is class number of $K$, $G(\chi)$ is the Gauss sum $\sum \chi(k) e^{2\pi i k /p}$. It is a non-trivial result that $G(\chi) = \sqrt{p}$ (when $p\equiv 1 \pmod{4}$). Some rearrangement gives $$\varepsilon^h = \frac{\sqrt p}{2^{(p-1)/2}}\prod_{k=1}^{(p-1)/2}\csc\frac{\pi k^2}p$$

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