On distribution of the number of semisimple rings of order at most x in an arithmetic progression

Abstract

Let \ell and q denote relatively prime positive integers. In this article, we derive the asymptotic formula for the summation \begin{align*} \sum_{n\leq x\atop n\equiv \ell \pmod q}S(n), \end{align*} where S(n) denotes the number of non-isomorphic finite semisimple rings with n elements.