In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average: $$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$ Based on Appell–Lerch sum properties and universal mock theta functions, we obtain the stronger version of the results of Bringmann and Lovejoy.

在本文中，我们将 $\overline {D}(a,M)$ 分解为模部分和模拟模部分，因此它直接给出了Bringmann 和Lovejoy 关于Maass 形式的著名结果。令 $\overline {p}(n)$是n 的分区数， $\overline {N}(a,M,n)$ 是n的超分区数，其秩与模M 一致。受 Hickerson 和 Mortenson 的启发，我们通过考虑秩与平均值的偏差，找到并证明了戴森秩的一般公式： $$ \begin{align*} \overline{D}(a,M) &=\sum\ limit_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n} . \end{对齐*} $$ 基于 Appell-Lerch 和属性和通用模拟 theta 函数，我们获得了Bringmann 和Lovejoy 结果的更强版本。