Dyadic Prime Product Representation of the Dirichlet Eta Function

This paper presents a new infinite product representation of the Dirichlet eta function, extending the Dyadic Prime Product formulation introduced by Odeh for the Riemann zeta function \cite{Odeh2025}. In this framework, $\zeta(s)$ is expressed as a double infinite product over primes and dyadic exponents, systematically separating the contribution of the even prime from that of the odd primes. Building on this foundation, we derive an analogous infinite product for $\eta(s)$ that embeds its alternating series nature into the dyadic structure. Through a sequence of analytical transformations, including geometric series expansions and Kronecker delta adjustments, we arrive at a unified expression that captures all prime contributions. Our main result is given by: \begin{equation}\eta(s) = \prod_{\substack{p \text{ prime}}} \prod_{k=0}^{\infty} \left[ 1 + p^{-s \cdot 2^{k}} - \delta_{p,2} \cdot \delta_{k,0} \cdot p^{1-s} - \delta_{p,2} \cdot \delta_{k,0} \cdot p^{-s \cdot 2^{k}} \cdot p^{1-s} \right]\end{equation} This formulation offers new structural insights into the eta function’s relation to prime distributions, enhancing our understanding of its analytic properties.