A New Expansion of the Reciprocal of the Riemann Zeta Function and Its Implications

We introduce a new analytic expansion for the reciprocal of the Riemann zeta function based on a Mittag–Leffler type decomposition associated with its nontrivial zeros. The construction yields explicit coefficient representations and convergence properties in regions of the complex plane excluding the poles induced by the zeros of the zeta function. We show that the uniform convergence of the proposed expansion in certain critical domains is equivalent to the Riemann Hypothesis. Numerical evaluations based on the first nontrivial zeros are provided to illustrate the behavior of the expansion and its convergence properties.