Formalization of $p$-adic $L$-functions in Lean 3

Abstract

The Euler-Riemann zeta function is a largely studied numbertheoretic object, and the birthplace of several conjectures, such as the Riemann Hypothesis. Different approaches are used to study it, including p-adic analysis : deriving information from p-adic zeta functions. A generalized version of p-adic zeta functions (Riemann zeta function) are p-adic L-functions (resp. Dirichlet L-functions). This paper describes formalization of p-adic L-functions in an interactive theorem prover Lean 3. Kubota-Leopoldt p-adic L-functions are meromorphic functions emerging from the special values they take at negative integers in terms of generalized Bernoulli numbers. They also take twisted values of the Dirichlet L-function at negative integers. This work has never been done before in any theorem prover. Our work is done with the support of mathlib 3, one of Lean's mathematical libraries. It required formalization of a lot of associated topics, such as Dirichlet characters, Bernoulli polynomials etc. We formalize these first, then the definition of a p-adic L-function in terms of an integral with respect to the Bernoulli measure, proving that they take the required values at negative integers. ACM Subject ClassificationMathematics of computing → Mathematical software; Theory of computation → Formal languages and automata theory Keywords and phrases formal math, algebraic number theory, Lean, mathlib Supplementary Material Copies of the source files relevant to this paper are available in a separate repository. Software: https://github.com/laughinggas/p-adic-L-functions Funding Ashvni Narayanan: EPSRC Grant EP/S021590/1 (UK) Acknowledgements The author is supported by EPSRC. The author would like to thank her PhD supervisor Prof Kevin Buzzard for several helpful insights. She would also like to thank Dr Filippo A. E. Nuccio for helpful conversations that helped give shape to the project. She is very grateful to the entire Lean community for their timely help and consistent support.