Irrationality of ℜLiₛ, Tiₛ, and Related Functions at Almost All Positive Integers

Let F:(0,∞)→ℝ be a real-valued function satisfying a polynomial reciprocal identity, and assume that its values at suitable small rational arguments satisfy a fixed-pattern linear-independence condition over ℚ. We establish a general criterion showing that F(n) is irrational for almost all positive integers n. More generally, the same conclusion holds along sequences of the form cnᵉ, where c is a fixed positive rational number and e is a positive integer. Using the linear-independence criterion of David, Hirata-Kohno, and Kawashima (DHK), we apply this result to inverse-tangent polylogarithms, polylogarithms at negative arguments, real principal-branch polylogarithms, and Legendre's chi function. We also establish quantitative upper bounds for the number of rational exceptions, joint linear-independence results, and estimates along geometric subsequences.