A Novel Explicit Formula for a Product Representation of a Chebyshev-Type Function with Zeros at Prime Powers

The distribution of prime numbers is a central problem in analytic number theory, intrinsically linked to the non-trivial zeros of the Riemann zeta function $\zeta(s)$. Explicit formulas, such as the Riemann-von Mangoldt formula, provide a crucial connection between sums over prime numbers and sums over these complex zeros. This paper derives a novel representation for a function, denoted $\hat{\Psi}(z)$, which is intrinsically related to the classical second Chebyshev function $\psi(x)$. This new representation manifests as a duality: one side is a product whose zeros precisely correspond to prime powers (specifically, at $z=\pm k\ln p$), while the other side is an exponential sum directly involving the imaginary parts of the non-trivial zeros of $\zeta(s)$. The derivation systematically employs fundamental tools from complex analysis, including Laplace transforms, Cauchy's Argument Principle applied to the completed zeta function $\xi(s)$, differentiation of generalized functions, and integral calculus. This novel form offers a unique analytical lens into the prime-zero duality, providing a compact and potentially computationally tractable framework for studying prime number distribution, exploring new avenues for analytical insights into the Riemann Hypothesis, and inspiring novel numerical algorithms for prime counting or zero localization.