Arithmetic properties of summands of partitions (Q1770968)

Let \(p(n)\) denote the number of (unrestricted) partitions of \(n\). In the paper under review the authors investigate the distribution of the summands of partitions of \(n\) in residue classes \(\bmod d\) for fixed \(d\geq 2\). They consider both the number and the sum of the parts in an arbitrary residue class \(\operatorname{mod} d\). Theorems 1.1 and 1.2 imply that, for almost all partitions of \(n\) (i.e., with the exception of \(o(p(n))\) partitions of \(n\) at most), the sum of the parts \(\equiv r\bmod d\) is \(n/d + O(n^{3/4}(\log n)^{1/2})\) which is sharp apart from the factor \((\log n)^{1/2}\). According to Theorem 1.3, with the abbreviation \(M=(6n)^{1/2}/(\pi d)\), the number of parts \(\equiv r \bmod d\) is \(\leq M\log M+Mc\) for \[ (\Gamma(r/d)^{-1}\int^\infty_{e^{-c}}t^{(r/d)-1}e^{-t}\, dt + o(1))p(n) \] partitions of \(n\) where \(\Gamma(s)=\int^\infty_0 t^{s-1}e^{-t}\,dt\) \((\text{Re\,}s>0)\). This implies that, for \(\omega(n)\nearrow\infty\), the number of parts \(\equiv r\bmod d\) is \((6n)^{1/2}(2\pi d)^{-1}\log n + O(n^{1/2}\omega(n))\) in almost all partitions of \(n\). This result is sharp as Theorem 1.4 shows.