| Summary: | 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 𝐿_{𝑝} norms of Littlewood polynomials
of degree 𝑛-1. We show that the arithmetic means of the Mahler’s
measure and the 𝐿_{𝑝} 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 𝑀_{𝛼} of
the 𝐿_{𝑝} 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 𝑓 defined by 𝑓(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.
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