On a method of Balasubramanian and Ramachandra (on the abelian group problem)
Let \(a(n)\) denote the number of non-isomorphic abelian groups with \(n\) elements, so that \(A(s) := \sum_{n=1}^\infty a(n)n^{-s} = \zeta(s)\zeta(2s)\zeta(3s)\ldots\) \((\text{Re} s>1)\), and further let \[ E(x) := \sum_{n\leq x}a(n) - \sum_{j=1}^{10}C_jx^{1/j}, \quad C_j = \prod_{k=1,k\not=j}^\infty \zeta\left({k\over j }\right) \] denote the error term in the asymptotic formula for the summatory function of \(a(n)\). The authors elaborate on the general method of \textit{R. Balasubramanian} and \textit{K. Ramachandra} [Hardy-Ramanujan J. 4, 13-40 (1981; Zbl 0483.10038)], specified to the function \(E(x)\). They prove that \[ \int_T^\infty E^2(u)u^{-4/3}e^{-u/X} \text{ d}u \gg \log^2T, \quad X = T^{200},\tag{1} \] \[ \min_{X\leq u\leq X^A}E(u)u^{-1/10} < -\exp\Biggl(B\biggl({\log X\over \log\log X}\biggr)^{1/2}\Biggr) \] with effective constants \(A, B > 0\), and an analogous lower bound result for ``max'' in place of ``min''. From (1) one can deduce an \(\Omega\)-result for \(E(u)\), but it should be remarked that essentially the same type of \(\Omega\)-result for \(E(u)\) follows from the bound \(\int_0^T E^2(u) \text{ d}u=\Omega(T^{4/3}\log T)\), proved earlier by the reviewer [\textit{A. Ivić}, J. Number Theory 27, 73-91 (1987; Zbl 0619.10040)].
number of finite non-isomorphic abelian groups
scientific article; zbMATH DE number 1089775
scientific article