Abstract
Let A be a set of positive integers with gcd (A) = 1, and let p A (n) be the partition function of A. Let c 0 = π√2/3. If A has lower asymptotic density α and upper asymptotic density β, then lim inf log p A (n)/c 0 √n ≧ √α and lim sup log p A (n)/c 0 √n ≦ √β. In particular, if A has asymptotic density α > 0, then log p A (n) ∼ c0√αn. Conversely, if α > 0 and log p A (n) ∼ c 0 √αn, then the set A has asymptotic density α.
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