Zeta-Regularization Connects Divergent Sums to Finite Scattering Phase Shifts — E8 Intelligence Research

**FINDING:** Zeta-regularization connects divergent sums (like \(1+2+3+\dots\)) to finite values (\(-1/12\)) via analytic continuation of the Riemann zeta function, and this technique is used in Lüscher-type formulas to extract infinite-volume scattering phase shifts from finite-volume lattice calculations. **MATH:** - Zeta-regularized sum: \(\sum_{n=1}^{\infty} n = \zeta(-1) = -\frac{1}{12}\) - Zeta-regularized product: \(\prod_{n=1}^{\infty} n = \sqrt{2\pi}\) (via \(\zeta'(0) = -\frac{1}{2}\ln(2\pi)\)) - Lüscher formula: Phase shift \(\delta(k)\) related to energy shift \(\Delta E\) in finite volume \(L\): \[ \Delta E = -\frac{2\pi}{mL^3} \sum_{\vec{n}} \frac{1}{k^2 - |\vec{n}|^2 (2\pi/L)^2} + \text{regularization} \] (Zeta-regularization tames the divergent sum over lattice momenta.) **CONNECTION:** - The value \(-1/12\) appears in the Casimir effect (parallel plates) and in the critical dimension of bosonic string theory (26 = 2 × 13, where 13 relates to the z Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com