Toupin, Daniel: The Shadow Euler Identity: A Family of Evaluations of the Completed Zeta Function at Glueball Celestial Weights via Products over the Riemann Zeros

Let ξ(s) = ½s(s−1)π^(−s/2)Γ(s/2)ζ(s) be the completed Riemann zeta function and let {γ_ρ} denote the positive imaginary parts of its nontrivial zeros. We first prove a universal product formula: for all s∈ℂ (under the Riemann Hypothesis), ξ(s)/ξ(½) = ∏_{γ_ρ>0} (1 + (s−½)²/γ_ρ²), expressing the ratio ξ(s)/ξ(½) as a product over the Riemann zeros with coupling a(s) = |s−½|. The formula has two regimes: for real s each factor exceeds 1 so ξ(s) ≥ ξ(½) (the completed zeta function is minimized at the