A Prime Power Equation

TL;DR

This paper introduces a new prime power equation derived from a special function related to the Riemann zeta function, providing a zero-free formulation and linking prime number theory with theta functions.

Contribution

It constructs a prime power equation free of zeta zeros using a novel function G and develops an infinite system of equations involving theta function values.

Findings

Derived a zero-free prime power equation from the zeta function.

Established an infinite system linking von Mangoldt's function with theta function values.

Connected prime number distribution with special functions and Fourier analysis.

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, $\theta_2(\tau)$.