AI 中文总结
本文沿用Zudilin与Zeilberger的相关策略，利用复轮廓积分等技术，将arctan(1/2)的无理数测度上界改进至8.585166。
AI 中文摘要
本文建立了arctan(1/2)无理数测度的新上界，即8.585166……。本文采用Zudilin和Zeilberger（他们改进了π的相关边界）的策略与证明技术，使用一族带有对称被积函数的复轮廓积分；通过p进赋值和专门定义的素数集合P_n推导了部分分式系数的整性与可除性，结合鞍点渐近分析，积分序列及其有理/对数分量的增长率给出了所需的无理数测度。
英文摘要
This paper establishes a new upper bound for the irrationality measure of arctan(1/2), namely 8.585166... Following the strategy and proof techniques of Zudilin and Zeilberger (who improved the bound for pi), we use a family of complex contour integrals with symmetric integrands. The integrality and divisibility of the partial-fraction coefficients are derived via p-adic valuations and a specially defined prime set P_n. Combined with saddle-point asymptotics, the growth rates of the integral sequence and its rational/logarithmic components yield the desired irrationality measure.
