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Asymptotic Density and the Asymptotics of Partition Functions

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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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  1. School of Mathematics Institute for Advanced Study, Princeton, NJ, 08540, U.S.A.

    M. B. Nathanson

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Nathanson, M.B. Asymptotic Density and the Asymptotics of Partition Functions. Acta Mathematica Hungarica 87, 179–195 (2000). https://doi.org/10.1023/A:1006789419031

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