Eta Derivative Symmetry and L'Hôpital Limits

The Zero-Weighted Eta function, $\tilde{\eta}(s) = \sum_{k=1}^{\infty} \frac{(-1)^{k-1}s^k}{k^s}$, admits a convergent Mellin–kernel representation and satisfies the Zero-Weighted Functional Equation (ZWF): $\frac{\tilde{\eta}(1-s),\eta(s)}{\tilde{\eta}(s),\eta(1-s)} = \frac{P(1-s)}{P(s)} , (2\pi)^{-s} , 2 \cos\Bigl(\frac{\pi s}{2}\Bigr)$. This formulation elucidates derivative-level symmetry in the Dirichlet eta functional equation $\eta(s) = \frac{\Gamma(1-s)}{\Gamma(s)} (2\pi)^{-s} 2 \cos\frac{\pi s}{2}, \eta(1-s)$, enabling systematic $m$-fold L’Hôpital analysis at zeros of arbitrary multiplicity. Consequently, the Zero-Weighted Eta function yields the Zero Derivative Equivalence: a $\Gamma$-independent structural constraint on derivatives at symmetric zeros, ensuring coinciding multiplicities and revealing total derivative symmetry.