An Investigation of the Rational Constant 5/2 Linking Apéry's Constant ζ(3) and the Euler-Mascheroni Constant γ via High-Precision Empirical Approximation

We report a high-precision empirical relationship connecting two fundamental constants of number theory and analysis: Apéry's constant ζ(3) and the Euler-Mascheroni constant γ: ζ(3) × e^( (1/γ) - 1 ) ≈ 5/2 While analytically non-exact (differing at the 4th decimal place, Δ ≈ 0.000497), this expression offers a remarkably compact functional approximation linking two historically decoupled constants. Furthermore, we contextualize this relation alongside the exact appearance of 5/2 as the leading rational coefficient in Apéry's classical accelerated series for ζ(3).