Let $n,a$ be fixed coprime positive integers. Let $A$ be the $n \times n$ matrix with entries $$A_{jk} = \csc\left(\dfrac{\pi(k-j)a}{n}\right)$$ for $1 \leq j \neq k \leq n$, and $A_{jj} = 0$ for $1 \leq j \leq n$. Would you know what is an upper bound for the spectral norm $\|A \|$of $A$? From a few experiments I have done, it seems that if $a < n$, then $\|A\| \leq n/a$. Is this true? Note that the case when $a = 1$ is well-known. See for instance Theorem 8.2, p.28, here: http://www.maths.lancs.ac.uk/~jameson/hilbert.pdf
Edit: I have now tried a few more computer experiments and it now seems (I made a mistake in the previous computations) that the spectral norm is at least very close to that when $a = 1$. This now makes sense: Since $a$ is coprime to $n$, the values $ja,ka \pmod{n}$ run over the entire set $\{1,\ldots,n\}$ and so the $a$ can be ignored. Sorry for the bad question!