On the differences between two kinds of mean value formulas of number-theoretic error terms

Abstract

In this paper, we shall consider differences between the "discrete" and the "continuous" mean value formulas for any number-theoretic error term. We first derive new type formulas of these differences, which are independent from the periodic Bernoulli function. As a direct consequence of this result, we shall give the alternative proofs of previous results in the case of the Dirichlet divisor and the circle problems, and improve the previous results of the differences in higher power cases. Furthermore, we apply our present result to the case of Rankin–Selberg's series.