Operator-valued zeta functions and Fourier analysis

TL;DR

This paper introduces an operator-valued approach to the Riemann zeta function, enabling analysis of its zeros using Fourier series without traditional analytic continuation methods.

Contribution

It proposes a novel operator-valued framework for the zeta function, allowing zero analysis through Fourier series in regions where the sum does not converge.

Findings

Locations of trivial zeros determined via Fourier series

Operator approach bypasses traditional analytic continuation

Potential new insights into the Riemann hypothesis

Abstract

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s= \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region ${\rm Re}\,s<1$ by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$.