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

The profound duality between the distribution of prime numbers and the non-trivial zeros of the Riemann zeta function, $\zeta(s)$, is a cornerstone of analytic number theory. This paper introduces a novel analytical framework, inspired by the principles of signal processing and systems theory, to derive a new product-sum identity that makes this duality explicit. Starting from the second Chebyshev function, $\psi(x)$, we employ a sequence of transforms, complex analysis techniques, and functional manipulations to construct a new function, $\hat{\Psi}(z)$. We prove that this function admits two distinct but equivalent representations: (i) an infinite product whose zeros are located precisely at the logarithmic prime powers, $z=\pm k\ln p$, and (ii) an exponential sum whose argument is explicitly determined by the ordinates of the non-trivial zeros of $\zeta(s)$. This identity provides a new analytical tool for studying the prime distribution and offers a fresh perspective on the spectral interpretation of the zeta zeros. The derivation serves as a case study in applying systems-theoretic thinking to classical problems in number theory, with potential implications for the development of new computational algorithms.