The L_4 norm of Littlewood polynomials derived from the Jacobi symbol

Abstract: 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}.