Constrained ternary integers

Abstract

An integer [Formula: see text] is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers [Formula: see text] with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the [Formula: see text]th (inverse) cyclotomic polynomials with ternary [Formula: see text]. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction [Formula: see text] of ternary integers.