On the average number of direct factors of a finite abelian group (Q1841254)

Let \(\tau(G)\) be the number of direct factors of a finite abelian group \(G\). The reviewer [\textit{E. Krätzel}, Acta Arith. 51, 369-379 (1988; Zbl 0633.10044)] proved \[ \sum_{|G|\leq x}\tau (G) =A_1 x\log x+A_2x +A_3\sqrt x\log x+A_4 \sqrt x+\Delta (x) \] with the estimation of the error term \[ \Delta(x)\ll x^{5/12} \log^4x. \] This result has been improved in the meantime by many authors. In this paper the new estimation \[ \Delta(x) \ll x^{47/130+ \varepsilon} \] is given. The new improvement follows from an estimation of a triple exponential sum, which was proved by \textit{H.-Q. Liu} and \textit{J. Wu} [Acta Arith. 89, 163-187 (1999; Zbl 0937.11038)] and is based on the work of \textit{P. Sargos} and \textit{J. Wu} [Acta Math. Hung. 87, 333-354 (2000; Zbl 0963.10045)] and \textit{E. Fouvry} and \textit{H. Iwaniec} [J. Number Theory 33, 311-333 (1989; Zbl 0687.10028)].