The irrationality measure of arctan1/2 is at most 8.585166

Abstract

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.