Mahler’s Conjecture on ξ(3/2) n mod 1

Abstract

Abstract K. Mahler’s conjecture: There exists no ξ ∈ ℝ+ such that the fractional parts {ξ(3/2) n } satisfy 0 ≤ {ξ(3/2) n } < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence xn = {ξ(3/2) n }, n =1, 2,... has asymptotic distribution function c 0(x), where c 0(x)=1 for x ∈ (0, 1].