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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.
Reviewer: A.Ivić (Beograd)

MSC:

11N45 Asymptotic results on counting functions for algebraic and topological structures
11N37 Asymptotic results on arithmetic functions
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References:

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