Krätzel, Ekkehard On the average number of direct factors of a finite abelian group. (English) Zbl 0633.10044 Acta Arith. 51, No. 4, 369-379 (1988). Let \(G\) be a finite abelian group. Let \(\tau(G)\) denote the number of direct factors, \(t(G)\) the number of unitary factors of \(G\) and \(T(x)=\sum \tau(G)\), \(T^ *(G)=\sum t(G)\), where the summation is extended over all abelian groups of order not exceeding \(x\). E. Cohen [Acta Arith. 6, 159–173 (1960; Zbl 0113.25305)] proved the representations \[ T(x)=\gamma _{1,1} x(\log x+2C-1)+\gamma _{1,2}x+\Delta (x), \]\[ T^ *(x)=c_{1,1} x(\log x+2C-1)+c_{1,2}x+\Delta ^ *(x) \] with \(\Delta (x)\ll \sqrt{x} \log ^ 2x\), \(\Delta ^ *(x)\ll \sqrt{x} \log x\). In this paper these results are improved by \[ \Delta (x)=\gamma _{2,1}\sqrt{x}( \log x+2C-1)+\gamma _{2,2}\sqrt{x}+O(x^{5/12} \log ^ 4x), \]\[ \Delta ^ *(x)=c_ 2\sqrt{x}+O(x^{11/29} \log ^ 2x). \] Both problems are connected with some divisor problems. In order to prove the estimate for \(\Delta(x)\) a new general asymptotic representation for \[ D(a;x)=\#\{(n_ 1,...,n_ 4): n_ 1,...,n_ 4\in {\mathbb N},\quad n_ 1^{a_ 1}\cdot...\cdot n_ 4^{a_ 4}\leq x\}, \] \(a=(a_ 1,a_ 2,a_ 3,a_ 4)\), \(1\leq a_ 1\leq a_ 2\leq a_ 3\leq a_ 4\), is stated, where the estimations are obtained by means of estimating double exponential sums. Reviewer: Ekkehard Krätzel (Jena) Cited in 6 ReviewsCited in 9 Documents MSC: 11N45 Asymptotic results on counting functions for algebraic and topological structures 11L07 Estimates on exponential sums 11N37 Asymptotic results on arithmetic functions Keywords:finite abelian group; number of direct factors; divisor problems; double exponential sums Citations:Zbl 0113.25305 PDFBibTeX XMLCite \textit{E. Krätzel}, Acta Arith. 51, No. 4, 369--379 (1988; Zbl 0633.10044) Full Text: DOI EuDML