On the Properties of the Möbius Function

Abstract

We formalized some basic properties of the Möbius function which is defined classically as μ(n) = 1, if n = 1; μ(n) = 0, if p² | n for some prime p; μ(n) = (−1)ʳ, if n = p₁p₂⋯pᵣ, where pᵢ are distinct primes, as e.g., its multiplicativity. To enable smooth reasoning about the sum of this number-theoretic function, we introduced an underlying many-sorted set indexed by the set of natural numbers. Its elements are just values of the Möbius function. The second part of the paper is devoted to the notion of the radical of number, i.e. the product of its all prime factors. The formalization (which is very much like the one developed in Isabelle proof assistant connected with Avigad’s formal proof of Prime Number Theorem) was done according to the book [13].