Research Coming Soon

All nontrivial zeros of the Dirichlet eta function $\eta(s)$, and hence of the Riemann zeta function $\zeta(s)$, are simple. This proof reveals a structural incompatibility between the Two-Factor Eta Equation, arising from $\eta(\rho)=0$, and the Eta Multiplicity Equation, derived from $\eta'(\rho)=0$. Both equations isolate the factor $2^{-\rho}$ and involve identical complex terms $(2m-1)^{-\rho}$ and $\zeta(\rho)$, but the latter includes real logarithmic weights. These weights systematically reduce the right-hand ratio in the Eta Mismatch Equation relative to the left-hand ratio, and this inequality persists in the infinite sum limit, precluding any multiplicity of zeta zeros. Thus, the mismatch approach establishes simplicity of all nontrivial zeros.