The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

Abstract

This paper proves that every zero of any n ^th, n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation ${f(x) + f(2x) + \cdots + f(nx) = 0, x \in \mathbb{R}}$. The continuity of the solutions depends on the sign of the real part of each zero.