On the number of parts of integer partitions lying in given residue classes (Q1682612)

This article studies properties of integer partitions of a given integer \(n\) whose parts satisfy certain properties. Namely, the authors determine an asymptotic formula \(\widehat{T}_{r,N}(n)\), which denotes the number of parts over all partitions of \(n\) which are congruent to \(r\) modulo \(N\). Earlier results involving this value relied on using techniques related to modular forms, but no such technique can be used for the generating function for \(\widehat{T}_{r,N}(n)\) itself. Instead, the authors use \textit{E. M. Wright}'s circle method [Q. J. Math., Oxf. II. Ser. 22, 107--116 (1971; Zbl 0215.33601)] and the Euler-Maclaurin summation formula to obtain their result.