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

David W. Farmer|arXiv (Cornell University)|Nov 21, 2022

Random Matrices and Applications5 citations

TL;DR

This paper argues that there is no compelling evidence to doubt the Riemann Hypothesis (RH), countering common misconceptions drawn from computational data. By analyzing the zeta function's behavior at extreme heights using random matrix theory, characteristic polynomials, and the concept of carrier waves, the author demonstrates that large values of ζ(1/2 + it) arise naturally from fluctuations in zero density, not from violations of RH. The key contribution is a unified framework explaining why computational anomalies do not undermine RH.

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.

Motivation & Objective

• To refute published arguments suggesting the Riemann Hypothesis might be false, particularly those based on computational observations of large zeta values.
• To clarify widespread misconceptions in the literature about computations of the zeta function, especially regarding Gram points and interval types.
• To develop a deeper understanding of the zeta function’s behavior at extreme heights, where computation fails, using random matrix theory and characteristic polynomials.
• To explain why large values of |ζ(1/2 + it)| are not surprising or indicative of RH failure, but instead arise from statistical fluctuations in zero density.
• To establish a conceptual framework—centered on carrier waves and density waves—linking the distribution of zeros to the size of the zeta function.

Proposed method

• Analyzing the zeta function’s value distribution via random matrix theory, particularly comparing ζ(1/2 + it) to characteristic polynomials of large random unitary matrices.
• Introducing the concept of 'carrier waves'—fluctuations in zero density that modulate the amplitude of the zeta function at large heights.
• Using numerical data from Haar-random matrices in U(N) for N = 62, 250, and 1000 to model behavior beyond computational reach.
• Applying the Riemann–Siegel formula and Weil’s explicit formula to connect zero distribution to function values and detect anomalies.
• Distinguishing between 'good' and 'bad' Gram points and showing their distinction vanishes at large N, undermining their use in testing RH.
• Surveying connections between L-functions and random matrix eigenvalues, emphasizing emergent statistical regularities over individual zero behavior.

Experimental results

Research questions

• RQ1 Why do computational results suggesting large values of |ζ(1/2 + it)| lead some to doubt the Riemann Hypothesis, despite no theoretical basis for such doubt?
• RQ2 How do carrier waves—fluctuations in the density of zeros—explain the observed large values of the zeta function at extreme heights?
• RQ3 To what extent do random matrix models accurately predict the statistical behavior of the zeta function beyond the range of direct computation?
• RQ4 Why are distinctions like 'good' vs. 'bad' Gram points misleading when assessing the validity of the Riemann Hypothesis?
• RQ5 What explains the apparent discrepancy between small-matrix simulations and theoretical predictions in the distribution of interval types around displaced Gram points?

Key findings

• The distinction between 'good' and 'bad' Gram points becomes statistically meaningless at large N, as shown by data from N = 250 and N = 1000 random matrices, where both types become equally likely.
• For N = 62, the ratio of Type II to Type I intervals at displaced Gram points is significantly different from 1, but this discrepancy diminishes as N increases, indicating finite-size effects.
• The proportion of Type I and Type II intervals converges to 1 across all displacement levels (−0.2δ to 0.2δ) as N increases, supporting the idea that local spacing patterns are universal.
• The data from random matrices show that larger gaps and clusters of zeros are more likely than predicted by Poisson-like models, consistent with the presence of secondary terms in the distribution.
• Carrier waves—arising from fluctuations in zero density—explain why |ζ(1/2 + it)| can be large without violating the Riemann Hypothesis, as these are statistical fluctuations, not structural anomalies.
• The behavior of S(t), the imaginary part of log ζ(1/2 + it), is correlated with the size of |ζ(1/2 + it)|: in regions of large values, S(t) tends to decrease, a phenomenon explained by the interplay of carrier and density waves.