Jacobsthal Function Bound Analysis

**Jacobsthal Function Bound Analysis** This paper presents an original refinement of upper bounds for the Jacobsthal function g(n), which describes the longest possible gap between consecutive integers all divisible by primes from a given set. Using methods from combinatorial number theory, residue classes, and probabilistic modeling, we derive a new general upper bound of g(n) ≤ φ(n) log(n)^2. These improvements lay groundwork for further explorations in prime gaps and modular sieving theory.**雅可比塔尔函数界值分析** 本文针对雅可比塔尔函数（Jacobsthal function）g(n)的上界提出了原创性精细化改进。该函数用于刻画给定素数集合中，所有可被该集合内素数整除的连续整数之间的最大可能间隔。借助组合数论、剩余类及概率建模的相关方法，我们推导出了g(n) ≤ φ(n) log²(n) 的全新通用上界。此项研究进展为素数间隔与模筛理论的后续探索奠定了坚实基础。