On a certain sum in number theory. II

Abstract:

We derive “exact order” of the function \[ \sum \limits _{k \leq \sqrt x } {{k^\rho }{{\min }^\beta }\left ( {\frac {{\sqrt x }}{k},\frac {1}{{{P_k}}}} \right )}.\] Here $\rho$ and $\beta$ are nonnegative real numbers and, for given real ${\delta _1},{\delta _2}, \cdots ,{\delta _r},{P_k} = {\max _j}\langle k{\delta _j}\rangle$ where $\langle t\rangle$, for real t, denotes distance of t from the nearest integer. Using our results, we obtain the solution of the basic problem in the theory of lattice points with weight in rational many-dimensional ellipsoids.