zeta_zero
plain-language theorem explainer
The arithmetic zeta function evaluates to zero at input zero. Number theorists using Dirichlet series or arithmetic function identities would cite this evaluation as a base case. The proof is a direct term simplification that unfolds the definition and reduces the conditional expression.
Claim. The arithmetic zeta function satisfies $ζ(0) = 0$.
background
This module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. The zeta function is introduced as the constant-1 function on positive integers via the abbreviation zeta := ArithmeticFunction.zeta. Upstream results supply the multiplicativity of J-automorphisms and the definition of zeta_apply, which encodes the piecewise behavior at zero.
proof idea
The proof is a one-line term-mode simplification. It unfolds the zeta abbreviation, applies ArithmeticFunction.zeta_apply, and reduces the resulting ite expression to the zero case.
why it matters
This supplies the base evaluation for the arithmetic zeta function inside the NumberTheory.Primes.ArithmeticFunctions module. It supports later Dirichlet inversion and prime-counting arguments in the Recognition framework, though it currently has no direct downstream citations. The result is independent of the J-cost, phi-ladder, and T0-T8 forcing chain.
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