  Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a
Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can
grow with n. We consider such polynomials for odd square-free n, where \phi(n)
coefficients are determined by the Jacobi symbol, but the remaining
coefficients can be freely chosen. When n is prime, these polynomials have the
smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2
among all Littlewood polynomials, namely (7/6)^{1/4}. When n is not prime, our
results show that the normalised L_4 norm varies considerably according to the
free choices of the coefficients and can even grow without bound. However, by
suitably choosing these coefficients, the limit of the normalised L_4 norm can
be made as small as the best known value (7/6)^{1/4}.
