The<i>L</i><sub>4</sub>norm of Littlewood polynomials derived from the Jacobi symbol
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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 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 normalized 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 normalized 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 normalized L 4 norm can be made as small as the best published value (7/6) 1/4 .
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