On the average number of unitary factors of finite abelian groups
\textit{E. Cohen} [Acta Arith. 6, 159--173 (1960; Zbl 0113.25305)] considered the monoid \(X\) of isomorphism classes of finite abelian groups with multiplication defined by the direct product. He called \(A\in X\) a unitary factor of \(G\in X\) if, for some \(B\in X\), we have \(AB=G\), and \(A\) and \(B\) are relatively prime, i.e. have no non-trivial common divisor in \(X\). Let \(t(G)\) denote the number of such factors. For example \(t(C(2)\times C(2))=3\) as we have one non-trivial and two trivial decompositions. Let \[ T^*(x) = \sum_{G\in X, |G| \leq x} t(G). \] Cohen has shown an estimate of the form \[ T^*(x) = c_1 x \log x + c_2 x + O(\sqrt{x} \log x). \] \textit{E. Krätzel} [Acta Arith. 51, No. 4, 369--379 (1988; Zbl 0633.10044)] has refined this to \[ T^*(x) = c_1 x \log x + c_2 x + c_3 \sqrt{x} + \Delta^*(x) \] for some \(\Delta^*(x)=O(x^{11/29} \log^2 x)\). For positive integers \(k,n\) let \(t_k(n)\) be such that \(\sum_{n=1}^\infty t_k(n)n^{-s}=\prod_{j=k}^\infty\zeta^{1+\varepsilon(j)}(js)\), where \(\zeta(s)\) is Riemann's zeta function and \(\varepsilon(j)\) equals \(1\) for odd \(j\) and \(0\) for even \(j\). Krätzel's approach was to express \(T^*(x)\) in terms of \(t_3(n)\) and the function \[ D(1,1,2;x) = \sum_{n_1 n_2 n_3^2 \leq 1} 1. \] The function \(D(1,1,2;x)\) is a sum of a simple main term, a ''middle term'' involving some exponential sums, and a remainder of the order \(O(x^{1/4})\). Krätzel's proof shows that if the middle term is \(O(x^{\theta+\varepsilon})\) for some \(\theta\geq 1/4\), then so is \(\Delta^*(x)\). The present author has given an improved upper bound for the middle term obtaining \(\theta = 63/178 < 0.35394\).
scientific article; zbMATH DE number 5798750