A Prime Power Equation

Abstract

A real valued function, $G$, is provided whose Fourier transform, $\hat G$, is an entire function that satisfies, $E(s)\zeta(s) = \hat G(\frac{s -\frac{1}{2}}{i})$. Then $\hat G(\gamma) = 0$ for all nonreal zeros, $\rho = \frac{1}{2} + i \gamma$, of $\zeta(s)$. Combined with Guinand's explicit formula we obtain a prime power equation free of zeta zeros. Using infinitely many translates of G, an infinite system of equations, indexed on the natural numbers, is obtained. The solution vector of this system is the vector of values of von Mangoldt's function, $\Lambda(n), n = 1, 2 \cdots$. The entries of the matrix are special values of the fourth power of the Jacobi theta function, $\theta_2(\tau)$.