The L_4 norm of Littlewood polynomials derived from the Jacobi symbol

Littlewood raised the question of how slowly the L4 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 φ(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 published asymptotic value of the normalised L4 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 L4 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 L4 norm can be made as small as the best published value (7/6) 1/4 .