Dominant residue classes concerning the summands of partitions

Abstract

For $d \leqq n^{1/8-\varepsilon}$, we determine in a large range of integers $N_1,\ldots,N_d$ the asymptotic number of partitions of $n$ with exactly $N_r$ parts congruent to r modulo $d$ for $1 \le r \le d$. In the second part of the paper we derive many results on the distributions of the parts in residue classes. In particular we obtain for $1 \leqq a < b \leqq d \leqq n^{1/8-\varepsilon}$, an asymptotic formula for the number of partitions of $n$ in which there are more parts $\equiv a (mod d)$ than parts $\equiv b (mod d)$.