GCD Matrices Associated with Nathanson’s Totient Function

Abstract

 Let ????(????) denote Nathanson’s Euler-type function for subsets of {1,2,…,????}, defined as the number of nonempty subsets ???? satisfying gcd(gcd????,????)=1. We introduce the second Möbius transform ????=????∗????, derive explicit formulas for ????, prove ????(????)>0 for every positive integer ????, and obtain lower bounds and asymptotic estimates. We then study the Nathanson GCD matrix ????????(????)=[????(gcd(????????,????????))]????,????=1????associated with a finite set ???? of distinct positive integers. Using divisor-incidence factorization, we prove positive definiteness for arbitrary finite sets, derive exact determinant formulas, and obtain an explicit inverse formula on factor-closed sets.