English
For a commutative group G, the product-Möbius inversion form is obtained from the additive version applied to G via the change of sign, yielding the equivalence between divisor products and antidiagonal products with μ-exponents.
Русский
Для коммутативной группы G формула инверсии по делителям через произведения на μ-степенях эквивалентна сумме через антидиагональ.
LaTeX
$$$\forall f,g : \mathbb{N} \to G\ (s:Set\mathbb{N}) (hs : ∀ m n, m \mid n → n ∈ s → m ∈ s) : (∀ n > 0, n ∈ s → (\prod i ∈ n.divisors, f i) = g n) \\iff \\ (∀ n > 0, n ∈ s → (\prod x ∈ n.divisorsAntidiagonal, g x.snd ^ μ x.fst) = f n).$$
Lean4
theorem bernoulli'_spec (n : ℕ) : (∑ k ∈ range n.succ, (n.choose (n - k) : ℚ) / (n - k + 1) * bernoulli' k) = 1 :=
by
rw [sum_range_succ_comm, bernoulli'_def n, tsub_self, choose_zero_right, sub_self, zero_add, div_one, cast_one,
one_mul, sub_add, ← sum_sub_distrib, ← sub_eq_zero, sub_sub_cancel_left, neg_eq_zero]
exact Finset.sum_eq_zero (fun x hx => by rw [choose_symm (le_of_lt (mem_range.1 hx)), sub_self])
