Asymptotic Estimates for Phi Functions for Subsets of { M + 1 , M + 2 , . . . , N }

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 fk(m,n) and Φk(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 fk(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 Φk(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. We shall obtain simple explicit formulas and asymptotic estimates for the four functions. 1The work of M.B.N. was supported in part by grants from the NSA Mathematical Sciences Program and the PSC-CUNY Research Award Program. 2Actually, 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. 2 INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (2007), #A54 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. Theorem 1. For nonnegative integers m < n, let f(m,n) denote the number of relatively prime subsets of {m + 1,m + 2, . . . , n}. Then f(m,n) = n ∑ d=1 μ(d) ( 2[n/d]−[m/d] − 1 ) and 0 ≤ 2n−m − 2[n/2]−[m/2] − f(m,n) ≤ 2n2[(n−m)/3]. Proof. El Bachraoui [1] proved that