Dyadic Prime Product: A New Infinite Product Representation of the Riemann Zeta Function and Its Verification

The Riemann zeta function, a cornerstone of analytic number theory, has traditionally been expressed through the Euler productformula, revealing its deep connection to prime numbers. In this work, we introduce a new infinite product representation ofthe zeta function, termed the Dyadic Prime Product, which systematically extends infinite product formulations beyond individualprime bases. By selecting a minimal subset of terms that are not products of previous or other terms, we construct an expansion that naturally accommodates both prime and composite numbers.At the core of this formulation lies the following double infinite product, termed the Dyadic Prime Product: \begin{equation}\zeta(s) = \prod_{p \text{ prime}} \prod_{k=0}^{\infty} (1 + p^{-s 2^k}),\end{equation} The Dyadic Prime Product is another representation of the Euler product formula for the Riemann zeta function, with thekey distinction that the product appears in the numerator rather than the denominator, offering an alternative perspective on prime distributions.This formulation elegantly captures all numbers through structured interactions of dyadic prime terms. We rigorously verify thecorrectness of this representation through step-by-step expansions, demonstrating that it reconstructs the full Dirichlet series of the zeta function. Additionally, an alternative proof establishes its equivalence to the Euler prime product by leveraging the geometric series identity, ultimately leading to a fundamental identity confirming the validity of the formulation.This new perspective not only reinforces the structure of prime-based infinite products but also provides deeper insight into the fundamental decomposition of numbers. The Dyadic Prime Product, in the form of the double infinite prime product, emerges as apowerful alternative framework for understanding the analytic and arithmetic properties of the Riemann zeta function.