The general divisor problem
Let \(a_ 1,a_ 2,...,a_ k\) be fixed integers with \(1\leq a_ 1\leq a_ 2\leq...\leq a_ k\) and let \(d(a_ 1,a_ 2,...,a_ k;n)\) denote the number of representations of an integer \(n\geq 1\) by \(n=n_ 1^{a_ 1} n_ 2^{a_ 2}... n_ k^{a_ k}.\) Further, let \(\Delta_ k(x)=\Delta (a_ 1,a_ 2,...,a_ k;x)\) be the error term in the asymptotic formula for the summatory function of \(d(a_ 1,a_ 2,...,a_ n;n).\) The estimation of \[ \alpha_ k=\inf \{\alpha \geq 0: \Delta_ k(x)\ll x^{\alpha}\} \] is, in general, very difficult. Therefore, the author considers in his paper the quantity \[ \beta_ k=\inf \{\beta \geq 0: \int^{X}_{1}\Delta^ 2_ k(x) dx\ll X^{1+2\beta}\}. \] The main result is: Let r be the largest integer which satisfies \((r-2)ar\leq a_ 1+a_ 2+...+a_{r-1}\) for \(2\leq r\leq k\), then \[ \beta_ k\geq g_ k=(r-1)/2(a_ 1+a_ 2+...+a_ r). \] Moreover, assuming the Lindelöf hypothesis to be true, then \(\beta_ k=g_ k\). Some more precise results are proved. An interesting application is given to the arithmetic function a(n), which denotes the number of non-isomorphic abelian groups of order n. If \(\Delta\) (x) denotes the error term in the asymptotic formula for \(\sum_{n\leq x}a(n)\), then it is proved that \[ \int^{X}_{1}\Delta^ 2(x) dx=\Omega (X^{4/3} \log X). \] A corollary of this result is \(\Delta (x)=\Omega (x^{1/6} \log^{1/2}x)\), which was proved by \textit{R. Balasubramanian} and \textit{K. Ramachandra} [Hardy-Ramanujan J. 4, 13-40 (1981; Zbl 0483.10038)].
number of non-isomorphic abelian groups of order n
scientific article; zbMATH DE number 4004288
scientific article