Ping Kuang

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

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...

Explicit Upper and Lower Bounds for S(x) and the O(x log log x) Order of the Error‑Term

This paper investigates the summatory divisor-sum function \(S(x)=\sum_{n\le x}\sigma(n)\). We analyze its oscillation induced by divisor distribution and adopt the maximum divisor-sum \(\sigma_{\max}(x+1)\). We establish an average-value formula and unconditional explicit bounds...