Asymptotic density and the asymptotics of partition functions ∗

Let A be a set of positive integers with gcd(A) = 1, and let pA(n) be the partition function of A. Let c0 = π √ 2/3. If A has lower asymptotic density α and upper asymptotic density β, then lim inf log pAn/c0 √ n ≥ √ α and lim sup log pA(n)/c0 √ n ≤ √ β. In particular, if A has asymptotic density α > 0, then log pA(n) ∼ c0 √ αn. Conversely, if α > 0 and log pA(n) ∼ c0 √ αn, then the set A has asymptotic density α. 1 The growth of pA(n) Let A be a nonempty set of positive integers. The counting function A(x) of the set A counts the number of positive elements of A that do not exceed x. Then 0 ≤ A(x) ≤ x, and so 0 ≤ A(x)/x ≤ 1 for all x. The lower asymptotic density of A is dL(A) = lim inf x→∞ A(x) x . The upper asymptotic density of A is dU (A) = lim sup x→∞ A(x) x . We have 0 ≤ dL(A) ≤ dU (A) ≤ 1 for every set A. If dL(A) = dU (A), then the limit d(A) = lim x→∞ A(x) x 1991 Mathematics Subject Classification. Primary 11P72; Secondary 11P81, 11B82,11B05.