Asymptotic estimates for phi functions for subsets of {m

Let f (m, n) denote the number of relatively prime subsets of {m+ 1, m + 2,. .. , n}, and let Φ(m, n) denote the number of subsets A of {m + 1, m + 2,. .. , n} such that gcd(A) is relatively prime to n. Let f k (m, n) and Φ k (m, n) be the analogous counting functions restricted to sets of cardinality k. Simple explicit formulas and asymptotic estimates are obtained for these four functions. A nonempty set A of integers is called relatively prime if gcd(A) = 1. Let f (n) denote the number of nonempty relatively prime subsets of {1, 2,. .. , n} and, for k ≥ 1, let f k (n) denote the number of relatively prime subsets of {1, 2,. .. , n} of cardinality k. Euler's phi function ϕ(n) counts the number of positive integers a in the set {1, 2,. .. , n} such that a is relatively prime to n. The Phi function Φ(n) counts the number of nonempty subsets A of the set {1,. .. , n} such that gcd(A) is relatively prime to n or, equivalently, such that A ∪ {n} is relatively prime. For every positive integer k, the function Φ k (n) counts the number of sets A ⊆ {1,. .. , n} such that card(A) = k and gcd(A) is relatively prime to n. Nathanson [2] introduced these four functions for subsets of {1, 2,. .. , n}, and El Bachraoui [1] generalized them to subsets of the set {m + 1, m + 2,. .. , n} for arbitrary nonnegative integers m < n. 1 We shall obtain simple explicit formulas and asymptotic estimates for the four functions. For every real number x, we denote by [x] the greatest integer not exceeding x. We often use the elementary inequality [x] − [y] ≤ [x − y] + 1 for all x, y ∈ R. n d=1 µ(d) 2 [n/d]−[m/d] − 1 and 0 ≤ 2 n−m − 2 [n/2]−[m/2] − f (m, n) ≤ 2n2 [(n−m)/3]. 1 Actually, our function f (m, n) is El Bachraoui's function f (m + 1, n), and similarly for the other three functions. This small change yields formulas that are more symmetric and pleasing esthetically.