AVERAGE MAHLER’S MEASURE AND Lp NORMS OF LITTLEWOOD POLYNOMIALS

Littlewood polynomials are polynomials with each of their coefficients in the set {−1, 1}. We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1. We show that the arithmetic means of the Mahler’s measure and the Lp norms of Littlewood polynomials of degree n − 1 are asymptotically e−γ/2 √ n and Γ(1+ p/2)1/p √ n, respectively, as n grows large. Here γ is Euler’s constant. We also compute asymptotic formulas for the power means Mα of the Lp norms of Littlewood polynomials of degree n− 1 for any p > 0 and α > 0. We are able to compute asymptotic formulas for the geometric means of the Mahler’s measure of the “truncated” Littlewood polynomials f̂ defined by f̂(z) := min{|f(z)|, 1/n} associated with Littlewood polynomials f of degree n − 1. These “truncated” Littlewood polynomials have the same limiting distribution functions as the Littlewood polynomials. Analogous results for the unimodular polynomials, i.e., with complex coefficients of modulus 1, were proved before. Our results for Littlewood polynomials were expected for a long time but looked beyond reach, as a result of Fielding known for means of unimodular polynomials was not available for means of Littlewood polynomials.