Explicit Upper and Lower Bounds for D(x) and the Error Term of Order O(x^ε)

Abstract

This paper studies the summatory divisor counting function D(x)=sum_{n≤x}τ(n). We derive its average‑value formula and establish explicit upper and lower bounds for D(x). The error term is governed by the maximum divisor count and satisfies the bound O(x^ε) for arbitrarily small positive ε. Unlike general summatory functions constrained by the classical Ω lower bound theorem, this function exhibits a constrained oscillation structure and thus falls outside the scope of the theorem. These findings provide a new analytical approach to the divisor problem and lattice‑point counting problem.