An Elementary Method for Estimating Error Terms in Additive Number Theory
Elmer K. Hayashi · Proceedings of the American Mathematical Society · 1975
Let ${R_k}(n)$ denote the number of ways of representing the integers not exceeding $n$ as the sum of $k$ members of a given sequence of nonnegative integers. Using only elementary methods, we prove a general theorem from which we deduce that, for every $\epsilon > 0$, \[ {R_k}(n) - c{n^\beta } e o({n^{\beta (1 - \beta )(1 - 1/k)/(1 - \beta + \beta /k) - \epsilon }})\] where $c$ is a positive constant and $0 < \beta < 1$.