Currently there are no reasons to doubt the Riemann Hypothesis

Should one believe the Riemann Hypothesis (RH)? Since it is a conjecture with no proposed roadmap to prove it, one point of view is that it should neither be believed nor disbelieved. Yet many mathematicians have an opinion, presumably backed up by logical reasoning.

Here we consider all published arguments for doubting RH. A paper of Ivić [57, 58] lists 4 reasons, and a paper of Blanc [12] provides a 5th reason. Three of those reasons involve speculation about the distribution of zeros and their relationship to the value distribution of the \(\zeta\)-function. A fundamental question is: what does the \(\zeta\)-function look like in a neighborhood of its largest values? The majority of this paper is a survey of prior results, which lead to an answer to that question.

We do not include any detailed proofs. The reason is that our goal is to provide intuition and to be persuasive. The arguments against RH generally take the form “It would be surprising if \(\mathbf{X}\text{.}\)” So, the burden we bear in refuting that argument is to give good reasons why \(\mathbf{X}\) is not surprising. To crystallize our main points we present 29 Principles which we hope are also useful for future reference.

Is the purpose of this paper to persuade that RH is true? Certainly not. But perhaps those who continue to doubt RH will realize that their belief is not based on good evidence. As for those who believe RH: perhaps someone will write a companion paper: Currently there are no good reasons to believe the Riemann Hypothesis. Note the subtle difference from the opposite of the title to this paper. Also useful would be a paper explaining why every currently known equivalence to RH is unlikely to be helpful for proving RH (as in [36]).

The \(\zeta\)-function is the simplest example of an L-function. All L-functions have properties similar to the \(\zeta\)-function, and all L-functions have an analogue of the Riemann Hypothesis. As much as possible we try to discuss the \(\zeta\)-function in isolation, but in a few places it is necessary to expand our perspective. We attempt to keep this paper self-contained, providing definitions and background as needed.

In Section 2 we provide basic definitions and background. In Section 3 we describe four Mistaken Notions which appear repeatedly in discussions of computations of the \(\zeta\)-function, some of which play an important role in the claimed reasons to doubt RH. In Section 4 we describe the connection between the distribution of zeros and the size of the \(\zeta\)-function, introducing carrier waves as a way to separate local from long-range behavior. In Section 5 we briefly describe the connection between the \(\zeta\)-function and unitary polynomials, and in Section 6 we provide an historical account of the connections to Random Matrix Theory. This leads to Section 7, where we use large unitary matrices to illustrate phenomena which occur far outside the range in which we can compute the \(\zeta\)-function. By the end of Section 7 we have a good understanding of the “typical” large values of the \(\zeta\)-function, and the relationship between the carrier wave, the density wave, and \(S(t)\text{,}\) but it is not until Section 8 that we address the most extreme values. Finally in Section 9 we use information from the prior sections to refute the three arguments against RH based on the distribution of zeros and values of the \(\zeta\)-function, and for completeness in Section 10 we cite recent results to refute the other two arguments against RH. Finally, in Section 11 we use the Principles to explain why the Mistaken Notions of Section 3 are, in fact, mistaken.