On the Riemann Hypothesis and Hilbert's Tenth Problem

TL;DR

This paper reviews methods related to the Riemann hypothesis and explores their implications for the classification of related conjectures within the arithmetical hierarchy, particularly focusing on their formal logical status.

Contribution

It provides an overview of past approaches to proving the Riemann hypothesis as a $\Pi_1^0$ sentence and discusses attempts to classify the Elliott-Halberstam conjecture similarly.

Findings

Riemann hypothesis can be expressed as a $\Pi_1^0$ sentence

Past methods have been summarized for this classification

Initial attempts to classify the Elliott-Halberstam conjecture as $\Pi_1^0$

Abstract

The Riemann hypothesis is one of the most famous unresolved problems in modern mathematics. The discussion here will present an overview of past methods that prove the Riemann hypothesis is a $\Pi_1^0$ sentence. We also end with some attempts towards showing the Elliott-Halberstam conjecture is $\Pi_1^0$.