mobius_seventy
plain-language theorem explainer
The declaration establishes that the Möbius function evaluates to -1 at 70. Number theorists applying inclusion-exclusion or Dirichlet inversion would cite this concrete case for the product of three distinct primes. The proof reduces to a direct native computation once the arithmetic function definition is unfolded.
Claim. $μ(70) = -1$, where $μ$ is the Möbius function that vanishes on non-squarefree arguments and equals $(-1)^k$ on squarefree arguments with $k$ distinct prime factors.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function as an arithmetic function from natural numbers to integers. The upstream abbreviation defines mobius directly as ArithmeticFunction.moebius, inheriting the standard properties that μ(n) = 0 whenever a squared prime divides n. The local setting keeps statements minimal so that deeper Dirichlet algebra can be added once basic interfaces stabilize.
proof idea
The proof is a one-line wrapper that applies native_decide after the definition of mobius is unfolded at the concrete argument 70.
why it matters
This supplies a verified concrete value for a squarefree integer with three prime factors, supporting any later use of Möbius inversion within the arithmetic functions module. No downstream theorems are listed, so the result functions as a basic foothold rather than a link in a longer chain. It remains isolated from the Recognition Science forcing chain or phi-ladder constructions.
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