Average Mahler’s measure and 𝐿_{𝑝} norms of Littlewood polynomials

Abstract

Littlewood polynomials are polynomials with each of their coefficients in the set { − 1 , 1 } \{-1,1\} . We compute asymptotic formulas for the arithmetic mean values of the Mahler’s measure and the L p L_p norms of Littlewood polynomials of degree n − 1 n-1 . We show that the arithmetic means of the Mahler’s measure and the L p L_p norms of Littlewood polynomials of degree n − 1 n-1 are asymptotically e − γ / 2 n e^{-\gamma /2}\sqrt {n} and Γ ( 1 + p / 2 ) 1 / p n \Gamma (1+p/2)^{1/p}\sqrt {n} , respectively, as n n grows large. Here γ \gamma is Euler’s constant. We also compute asymptotic formulas for the power means M α M_{\alpha } of the L p L_p norms of Littlewood polynomials of degree n − 1 n-1 for any p > 0 p > 0 and α > 0 \alpha > 0 . We are able to compute asymptotic formulas for the geometric means of the Mahler’s measure of the “truncated” Littlewood polynomials f ^ \hat {f} defined by f ^ ( z ) := min { | f ( z ) | , 1 / n } \hat {f}(z) := \min \{|f(z)|,1/n\} associated with Littlewood polynomials f f of degree n − 1 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 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.