The saddle-point method for general partition functions (Q783664)

The saddle-point method has already proved to be useful in number theory, such as in deriving precise results for the \(y\)-smooth numbers [\textit{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 [\textit{T. Rivoal} and \textit{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.