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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 Φ = i+k=j+l qiq k qj q l satisfies Φ ≥ 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 Φ, (2) the study of the critical points of Φ,doi:10.5802/ambp.334 fatcat:l27s5h7ge5azlkpuwiysviwgbi