Uniform cubic bounds for irrational values of the dilogarithm

Abstract

We study explicit denominator conditions ensuring the irrationality of the dilogarithm at positive and negative rational points. We prove that Li_2(p/q) is irrational whenever q > (4e^{4}/27)p^3, and obtain an explicit, slightly smaller threshold for Li_2(−p/q). The positive argument specializes the Rhin–Viola integral family and uses a uniform estimate for a restricted prime divisor. The negative argument combines the same arithmetic construction with three adjacent integral moments and an explicit maximization. We also give finite-numerator refinements by interpolating certified partition bounds and by introducing a comparison measure with a quadratic tail. In particular, Li_2(5/q) is irrational for every integer q ≥ 418. The proof separates the established integral and determinant constructions from the parameter estimates and finite certificates used here.