The \(p^{a}\)-regular partition function modulo \(p^{j}\) (Q700199)

Let \(b_l(n)\) denote the number of \(l\)-regular partitions of the natural number \(n\), that is the number of partitions of \(n\) into parts not divisible by \(l\), or equivalently, the number of partitions of \(n\) such that no part occurs \(l\) or more times. Let \(p\) be a prime, \(a,j\) positive integers, \(l=p^a\), \(1\leq i\leq p^j-1\). The author obtains estimates for the number of integers, \(k\), such that \(k\leq X\) and \(b_l(k)\equiv i\pmod {p^j}\).