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The saddle-point method for general partition functions. (English) Zbl 1456.11198

The saddle-point method has already proved to be useful in number theory, such as in deriving precise results for the \(y\)-smooth numbers [E. Saias, J. Number Theory 32, No. 1, 78–99 (1989; Zbl 0676.10028)], or in studying arithmetic nature of the values of the Riemann zeta function at odd integers [T. Rivoal and W. Zudilin, Sémin. Lothar. Comb. 81, B81b, 13 p. (2020; Zbl 1470.11203)].
In the paper under review, the authors make use of the saddle-point method to get asymptotic formulas for the the number \(p_\Lambda(n)\) of partitions of \(n\) all of whose summands belong to \(\Lambda\), where the subset \(\Lambda \subset \mathbb{Z}_{\geqslant 1}\) satisfies some restricted conditions. The results are given in the form of a main term written as a full asymptotic series and an effective small error term.

MSC:

11P82 Analytic theory of partitions
05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
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