simp_rw [ArithmeticFunction.natCoe_apply, ArithmeticFunction.zeta_apply, ite_false, Nat.cast_one,
  mul_eq_zero, Nat.cast_ite, CharP.cast_eq_zero, mul_ite, mul_zero, true_and]
⊢ ↑ζ 1 = 1 ∧ ∀ (a b : ℕ), ↑ζ (a * b) = ↑ζ a * ↑ζ b
⊢ ∀ (a b : ℕ), (if a = 0 ∨ b = 0 then 0 else ↑1) = if b = 0 then 0 else (if a = 0 then 0 else ↑1) * ↑1