The L₄ norm of Littlewood polynomials derived from the Jacobi symbol

Jonathan Jedwab and Kai-Uwe Schmidt

Vol. 257 (2012), No. 2, 395–418

DOI: 10.2140/pjm.2012.257.395

Abstract

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 normalized 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 normalized 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 normalized L₄ norm can be made as small as the best published value (7∕6)^1∕4.

Keywords

Littlewood polynomial, norm, asymptotic, multiplicative character, merit factor

Mathematical Subject Classification 2010

Primary: 11B83, 11C08

Secondary: 94A55

Milestones

Received: 4 August 2011

Accepted: 26 August 2011

Published: 4 July 2012