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 \(S(x)=∑n≤ xσ(n)\). We analyze its oscillation induced by divisor distribution and adopt the maximum divisor-sum \(σₘₐₓ(x+1)\). We establish an average-value formula and unconditional explicit bounds for \(S(x)\). The leading term is \(π^2/12x²\), and the fluctuation is dominated by \(σₘₐₓ(x+1)\). With Robin’s unconditional bound, the error-term order \(O(xloglog x)\) is rigorously verified. The oscillation model and extremal-value analysis provide a new approach for the study of divisor-sum summatory functions.