  We investigate the problem of counting the real or complex Hadamard matrices
which are circulant, by using analytic methods. Our main observation is the
fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity
$\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with
equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled
circulant complex Hadamard matrix. This suggests three analytic problems,
namely: (1) the brute-force minimization of $\Phi$, (2) the study of the
critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We
explore here these questions, with some results and conjectures.
