An asymptotic formula for the logarithm of generalized partition functions.

Let Λ = { λ j } j = 1 ∞ be a set of positive integers, and let p Λ (n) be the number of ways of writing n as a sum of positive integers λ ∈ Λ such that χ (λ) = 1 and f (λ) ≡ j (mod m) , where m ≥ 1 and j ≥ 0 are fixed integers. Here, χ and f are a certain multiplicative function and an additive function, respectively. In this paper, we obtain the asymptotic formula for ln ∑ n ≤ x p Λ (n) ∼ ln p Λ ([ x ]) as x → ∞ . [ABSTRACT FROM AUTHOR]