Average Mahlers measure and L p L_p norms of Littlewood polynomials Academic Article

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 Mahlers 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 Mahlers measure and the L p L_p norms of Littlewood polynomials of degree n 1 n-1 are asymptotically e / 2 n e^{-gamma /2}sqrt {n} and ( 1 + p / 2 ) 1 / p n Gamma (1+p/2)^{1/p}sqrt {n} , respectively, as n n grows large. Here gamma is Eulers constant. We also compute asymptotic formulas for the power means M M_{alpha } of the L p L_p norms of Littlewood polynomials of degree n 1 n-1