Riemann Hypothesis Phase 1: Project Structure Public

Introduction

We have begun to see examples of collaborative math projects including PolyMath projects, Math Overflow, LMFDB and the use of blueprints and github in Lean formalization efforts such as Equational, PNT, and FLT. At the same time, we are beginning to see computational tools that can accelerate work: code generation for numerical experimentation, machine learning for finding extreme examples and test functions, automated proof assistants for resolving at least technical lemmas, LLMs for generating ideas and searching through papers for coincidences.

This presents a unique time to launch a large-scale collaboration to further understand and ultimately resolve the Riemann Hypothesis.

How to participate

You can Request Edit Access or reach out by Email with an introduction and how you would like to contribute.

We are not accepting any purported proofs or ideas yet, but you can start documenting these and merge them in during phases 2 and 3.

Math Context

There is 150 years’ worth of attempts, computations, and intuition, which can be hard to consolidate.

• Ivic explained reasons to doubt while Farmer explained why there is no reason to doubt.
• There’s significant numerical version but even at height \(10^{27}\) the discrepancy \(S(t)\) only reaches ~3.3 as shown by Bober and Hiary so the behavior of the zeta function is still very constrained. Odlyzko numerically verified at height \(10^{22}\) but remains agnostic.
• There are analogies to the solved Riemann Hypothesis for curves but the zeta functions there are rational functions.
• There have been attempts to find a Hilbert-Polya operator, further motivated by calculations from random matrix theory, lead to Bombieri and Garret’s work on Pseudo-Laplacians that ultimately was shown to be able to account for at most 94% of the zeros.
• RH has been continually generalized to broader universes of functions like the Selberg Class but we do not know a single function that matches just the analytic behavior of Riemann Zeta and provably satisfies the Riemann Hypothesis.
• The De-Bruijn Newman Constant is non-negative, which has been interpreted as saying that RH is “barely true”.
• It has been proven that >41% are on the critical line and 100% are arbitrarily close to the critical line, but the best known zero-free regions slowly converge to the 1-line.
• There’s simple formulations like showing that \(\sum_{n=1}^N \mu(n)\ll\sqrt{N}\) or \(\sum_{n=1}^N \lambda(n) \le 0\) (at least on average) but both conjectures were (non-constructively) disproven.

A lot of attempted directions ultimately reached a dead-end or provable obstruction, but lack of broad awareness of these can lead to repeated wasted effort. Thus there is value in collecting together known lines of attacks and known obstructions, to more efficiently explore new and existing directions.

Phase 1: Project Infrastructure

Set up the structure of the project: how to divide responsibilities, how to communicate and collaborate, how to welcome contributions from different backgrounds and skill sets, how to balance openness and privacy.
Understand what tooling needs to be built to support this, decide on a set of subteams.

Phase 2: Mapping the Problem Space

Documenting past attempts, relevant theory, history, heuristics, obstructions.
Ensuring the different formulations are properly formalized in Lean.
Finding reductions to search problems for special objects, that could be amenable to ML.

Phase 3: Full speed ahead

Execute on each track of the project.