# Rational approximations to the zeta function

## TL;DR

This paper introduces a sequence of rational functions that converge to the zeta function, with simple matrix-based numerators and denominators, offering a potential spectral approach to the Riemann hypothesis.

## Contribution

It presents a novel rational approximation framework for the zeta function, linking it to spectral problems and enabling potential quantitative analysis.

## Key findings

- Rational functions converge to the zeta function locally uniformly.
- Numerators and denominators are characteristic polynomials of simple matrices.
- The approach relates to spectral problems similar to Connes' analysis.

## Abstract

This article describes a sequence of rational functions which converges locally uniformly to the zeta function. The numerators (and denominators) of these rational functions can be expressed as characteristic polynomials of matrices that are on the face of it very simple. As a consequence, the Riemann hypothesis can be restated as what looks like a rather conventional spectral problem but which is related to the one found by Connes in his analysis of the zeta function. However the point here is that the rational approximations look to be susceptible of quantitative estimation.