Theorems · Theorem · number theory
ArithmeticFunction.IsMultiplicative.iff_ne_zero
∀ {R : Type u_1} [inst : MonoidWithZero R] {f : ArithmeticFunction R},
f.IsMultiplicative ↔ f 1 = 1 ∧ ∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → m.Coprime n → f (m * n) = f m * f nA recapitulation of the definition of multiplicative that is simpler for proofs
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MonoidWithZero
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroClass.zero_mulproof · cited by 1,625
- eq_or_neproof · cited by 1,117
- MonoidWithZerostatement and proof · cited by 456
- ArithmeticFunctionstatement and proof · cited by 290
- ArithmeticFunction.IsMultiplicativestatement · cited by 45
- ArithmeticFunction.map_zeroproof · cited by 34
Cited by5
Results whose statement or proof uses this declaration.
- ArithmeticFunction.isMultiplicative_zetaproof · cited by 5
- ArithmeticFunction.isMultiplicative_oneproof · cited by 2
- ArithmeticFunction.isMultiplicative_moebiusproof · cited by 2
- DirichletCharacter.isMultiplicative_toArithmeticFunctionproof · cited by 1
- ArithmeticFunction.IsMultiplicative.prodPrimeFactorsproof · cited by 1