Non-vanishing of Dirichlet series without Euler products

TL;DR

This paper presents a new proof demonstrating that the Riemann zeta function does not vanish in the half-plane where the real part exceeds 1, notably without relying on its Euler product representation.

Contribution

The paper introduces a novel proof technique for the non-vanishing of the Riemann zeta function that does not depend on the Euler product.

Findings

Proves non-vanishing of ζ(s) for σ > 1 without Euler product

Provides a new approach to understanding the zeta function's properties

Enhances methods for analyzing Dirichlet series

Abstract

We give a new proof that the Riemann zeta function is nonzero in the half-plane $\{s\in{\mathbb C}:\sigma>1\}$. A novel feature of this proof is that it makes no use of the Euler product for $\zeta(s)$.