Hurwitz Zeta Function

The Hurwitz Zeta Function is the special function ζ(s,a) that the book uses to give an exact value for the total energy of an infinite recursive harmonic series once that series converges. The total energy across all harmonics is E⊚ = E_0 sum(n=0 to infinity) (n + 1)^(2−γ), which converges only when 2 − γ < −1, that is γ > 3. For the case the book works through, γ = 3.5, the terms become (n + 1)^(−1.5) and the sum is the Hurwitz zeta function with arguments s = 1.5 and a = 1, which equals the Riemann zeta value ζ(1.5), giving a finite total of about 2.61 E_0. The book uses this as a concrete demonstration that an infinite harmonic series can be physically realizable, alongside the general condition γ > 3. The formal statement is E⊚ = E_0 ζ(1.5, 1) ≈ 2.612 E_0, where E⊚ is the total energy for γ = 3.5, E_0 the fundamental energy scale, ζ(s,a) the Hurwitz Zeta Function, s = 1.5 its first argument and a = 1 its second. Dimensions as the book gives them: E⊚ and E_0 [J]; ζ, s, a [∅]; the book's check is consistent. It connects to the Convergence Parameter, Amplitude Scaling, Energy Conservation and Spectral Density Function.