An Alternative Approach to 1 − γ and Riemann ζ Function

Abstract

Although the relationship ( $\sum_{k=2}^\infty \frac{\zeta(k) - 1}{k} = 1 - \gamma$ ) between the harmonic series, the logarithmic function, and the Riemann Zeta function is known, explicit elementary derivations are scarce in the literature. A novel derivation of this relationship (usually found in analytic number theory) is presented using only Taylor series expansions and telescoping sums, avoiding complex analysis and integral representations. This method provides a transparent connection between the harmonic series, the logarithmic function, and the Riemann Zeta function. All limit exchanges are analytically justified, convergence analysis is provided, and numerical validation demonstrates the practical utility of this approach. This elementary methodology offers pedagogical value and computational accessibility for both theoretical and applied contexts.