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