The distribution of values of the enumerating function of finite, non-isomorphic Abelian groups in short intervals (Q999104)

Let \(a(n)\) denote the number of finite non-isomorphic abelian groups, which is a multiplicative function satisfying \(a(p) = 1, a(p^2) = 2\) and \(a(n)\) is generated by \(\zeta(s)\zeta(2s)\zeta(3s)\ldots\), where \(\zeta(s)\) is the Riemann zeta-function. The reviewer [Math. Nachr. 101, 257--271 (1981; Zbl 0484.10028)] obtained an estimate for \[ A_k(x;h) = \sum_{x<n\leq x+h, a(n)=k}1, \] where \(k\geqslant1\) is fixed and \(h\) is ``short'' in the sense that \(h = o(x)\) as \(x\to\infty\). This was later improved by the author [Lattice Points. Mathematics and its applications: East European Series, 33. Dordrecht etc.: Kluwer Academic Publishers; Berlin: VEB Deutscher Verlag der Wissenschaften (1988; Zbl 0675.10031)], who connected the problem to the estimation of the error term in certain divisor problems (generated by the functions \(\zeta(s)\zeta(2s)\) and \(\zeta(s)\zeta(2s)\zeta(3s)\)). In the present paper he makes some further improvements in this topic by proving, in an elementary way, that \[ A_k(x;h) = (P_k+o(1))h + O(x^{\alpha_{1,2}+\varepsilon}). \] Here \(P_k \geqslant0\) is a well-defined constant, and \(\alpha_{1,2}\) is the constant related to a certain error term connected with \(\sum_{mn^2\leqslant x}1\) (generated by \(\zeta(s)\zeta(2s)\)) if \(\alpha_{1,2}< 1/2\). Finally he shows how the value \(\alpha_{1,2} = 369/1667 = 0.22135\ldots\,\) is permissible.