Estimates in the general divisor problem
Three theorems are proved, which involve upper bounds for \[ \Delta({\mathbf a};x)=\sum_{n_ 1^{a_ 1} n^{a_ 2}_ 2\cdots n^{a_ p}_ p\leq x} 1-\sum^ p_{i=1}\left(\prod^ p_{{j=1\atop j\neq i}}\zeta\bigl({a_ j\over a_ i}\bigr)\right)x^{1/a_ i} \] if \(1\leq a_ 1< a_ 2<\cdots< a_ p\) are given integers, and \(n_ j\) \((\geq 1)\) are integer variables. If some of the \(a_ j\)'s are equal, then the appropriate limits in the second sum above are to be taken. The function \(\Delta({\mathbf a}; x)\) may be considered as the error term in the general divisor problem, and there exists a considerable literature on this subject [see e.g. the reviewer's paper in J. Number Theory 27, 73-91 (1987; Zbl 0619.10046) for omega-results]. The author's first theorem says that, if \(p\geq 2\) and \((p-2)a_ p< 2a_ 1+ a_ 2+\cdots +a_{p- 1}\), then \[ \Delta({\mathbf a};x)\ll x^{(p-1)/(2a_ 1+a_ 2+\cdots +a_ p)}\log^ p x. \] The formulation of the remaining theorems is more complicated, and for this reason they will not be stated here. The proofs involve delicate transformations of multiple exponential sums and the use of the technique of exponent pairs.
transformations of multiple exponential sums
scientific article; zbMATH DE number 404315
scientific article