New estimates in the four-dimensional divisor problem with applications (Q624234)

This paper uses three-dimensional exponential sums to estimate the error terms for the summatory function of various arithmetic functions. Given positive integers \(a,b,c,d\) define the divisor function \(d(n;a,b,c,d)\) as the number of representations of \(n\) in the form \(n_1^an_2^bn_3^cn_4^d\). The paper then estimates the error term in the summatory function for this divisor function, under various conditions on the exponents \(a,b,c,d\). The first application concerns the error term for the sum of \(d_e(n)^2\), where \(d_e(n)\) is the number of ``exponential divisors'' of \(n\), that is to say products \(d=\prod_{p| n} p^{v(p,d)}\) for which each exponent \(v(p,d)\) divides the corresponding exponent \(v(p,n)\) occuring in \(n\). It is shown that the error term is of order \(O(x^{45/127}(\log x)^5)\). The second application concerns the number of \(4\)-th power-full numbers up to \(x\), for which an error term \(O(x^{35/317}(\log x)^5)\) is achieved. Finally the number of direct factors of distinct isomorphism classes of abelian groups of order at most \(x\) is investigated, and an error term \(O(x^{235/568}(\log x)^5)\) is obtained. The basic results are proved using triple exponential sums, estimated via a version of van der Corput's method. This uses lemmas from \textit{E. Krätzel} [Abh. Math. Sem. Univ. Hamburg, 62, 191--206 (1992; Zbl 0776.11057)] and from \textit{R. Seibold} and \textit{E. Krätzel} [Analysis, 18, 201--215 (1998; Zbl 0922.11069)].