Currently there are no reasons to doubt the Riemann Hypothesis: The zeta function beyond the realm of computation

TL;DR

This paper argues that doubts about the Riemann Hypothesis are unfounded by analyzing theoretical and computational evidence, emphasizing the role of random matrix theory and the behavior of the zeta function near its large values.

Contribution

It provides a comprehensive review and refutation of common misconceptions about the zeta function and the Riemann Hypothesis using advanced mathematical theories and illustrative examples.

Findings

Arguments claiming the hypothesis is false are not well-founded.

The behavior of the zeta function near large values can be understood through random matrix theory.

Computational evidence can be misleading due to complex fluctuations and carrier waves.

Abstract

We examine published arguments which suggest that the Riemann Hypothesis may not be true. In each case we provide evidence to explain why the claimed argument does not provide a good reason to doubt the Riemann Hypothesis. The evidence we cite involves a mixture of theorems in analytic number theory, theorems in random matrix theory, and illustrative examples involving the characteristic polynomials of random unitary matrices. Similar evidence is provided for four mistaken notions which appear repeatedly in the literature concerning computations of the zeta-function. A fundamental question which underlies some of the arguments is: what does the graph of the Riemann zeta-function look like in a neighborhood of its largest values? We explore that question in detail and provide a survey of results on the relationship between L-functions and the characteristic polynomials of random matrices. We highlight the key role played by the emergent phenomenon of carrier waves, which arise from fluctuations in the density of zeros. The main point of this paper is that it is possible to understand some aspects of the zeta function at large heights, but the computation evidence is misleading.