## Comparing the number of abelian groups and of semisimple rings of a given order.(English)Zbl 0848.11044

Let as usual $$a(n)$$ denote the number of non-isomorphic Abelian groups with $$n$$ elements, and $$S(n)$$ the number of semisimple rings with $$n$$ elements. These arithmetic functions are generated by $$\prod^\infty_{j= 1} \zeta(js)$$, $$\prod^\infty_{j= 1} \prod^\infty_{m= 1} \zeta(jm^2 s)$$, respectively, they are multiplicative and $$a(p^\alpha)= S(p^\alpha)$$ for primes $$p$$ and $$\alpha= 1, 2, 3$$. This motivates the author to study the summatory function of $$a(n)/S(n)$$. His main result is the asymptotic expansion $\sum_{n\leq x} {a(n)\over S(n)}= Ax+ x^{{1\over 4}} \sum^{M(x)}_{k= 0} A_k(\log x)^{- k- {7\over 6}}+ O(x^{{1\over 4}} \exp(- c(\log x)^{{3\over 5}}(\log \log x)^{- {1\over 5}})),$ where $$A> 0$$, $$M(x)= [c_0(\log x)^{{3\over 5}}(\log \log x)^{- {6\over 5}}]$$ and $$A_k\ll (bk)^k$$ with certain constants $$c_0$$, $$b> 0$$. The proof starts from the Dirichlet series representation $\sum^\infty_{n= 1} a(n)/(S(n) n^s)= \zeta(s) \zeta^{- {1\over 6}}(4s) U(s)$ $$(\text{Re } s> 1)$$, where $$U(s)$$ stands for a Dirichlet series that is absolutely convergent for $$\text{Re } s> 1/5$$. The author then uses convolution and complex integration, where considerable skill is displayed. The analysis is based on previous works of A. Selberg, J.-M. De Koninck and the reviewer, and W. G. Nowak. The author remarks that his proof can be generalized to an arbitrary $$r$$-th moment of $$a(n)/S(n)$$, where $$r> 0$$ is a fixed real number, and further generalizations are certainly possible.