A New Identity Relating Weighted Sums of Primes and Sums of Reciprocals of Riemann Zeros

In this paper we discover and prove a new identity that establishes an exact relationship between a certain weighted sum over prime numbers and the sum over nontrivial zeros of the Riemann zeta function of the reciprocal of ρ(1−ρ). By analyzing a convergent series involving the natural logarithm of integers, and applying the fundamental theorem of arithmetic, we expand this series into an infinite expression that involves only primes, thereby eliminating any explicit appearance of composite numbers. The resulting identity is presented in two equivalent forms—a single sum and a double sum—each of which equates a purely prime-dependent infinite series to a combination of elementary constants together with the sum over the zeros. The derivation does not rely on the Riemann Hypothesis; it uses only classical analytic tools and the fundamental theorem of arithmetic. Numerical computations confirm the correctness of the identity. We also discuss its conceptual kinship with the Riemann–Weil explicit formula, illustrating how the identity reflects a dual relation between primes and zeros in a concrete setting, without being a direct special case of that formula.