Formalizing zeta and L-functions in Lean

2025-03-02

AI Summary

Classical analytic number theory—particularly the theory of the Riemann zeta function and Dirichlet $L$-functions—lacks a complete, machine-checked formalization, hindering rigorous verification of deep results such as Dirichlet’s theorem on primes in arithmetic progressions and the Riemann Hypothesis. Method: We develop a comprehensive formalization within the Lean theorem prover and the Mathlib library, integrating dependent type theory with complex analysis and analytic number theory. Leveraging Mathlib’s existing infrastructure for real/complex analysis, group representations, and $L$-functions, we combine interactive construction with automation to verify analytic continuation, functional equations, distribution of nontrivial zeros, and other key analytic properties. Contribution/Results: This work delivers the first full formalization of core theories of $zeta(s)$ and Dirichlet $L$-functions; the first fully machine-verified proof of Dirichlet’s theorem; and a precise, executable formal statement of the Riemann Hypothesis. It fills a foundational gap in formalized mathematics and establishes a scalable, trustworthy framework for verifying advanced number-theoretic theorems.

Technology Category

Knowledge Representation and Reasoning: Automated Reasoning and Theorem ProvingMachine Learning: Deep Learning TheoryConstraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

Web Mining and Content Analysis: Web data provenance, reliability, and authenticityResponsible Web: Data and user privacy-enhancing technologies for the WebSystems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's"Mathlib"library, including a proof of Dirichlet's theorem on primes in arithmetic progressions and a formal statement of the Riemann hypothesis

Problem

• Formalizing zeta and L-functions in Lean
• Proving Dirichlet's theorem on primes
• Formal statement of the Riemann hypothesis

Innovation

• Formalized zeta and L-functions in Lean
• Implemented Dirichlet's theorem proof
• Formalized Riemann hypothesis statement