Theorems · Theorem · number theory
Summable.norm_lt_one
∀ {F : Type u_1} [inst : NormedDivisionRing F] [CompleteSpace F] {f : ℕ →* F},
Summable ⇑f → ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1If f is multiplicative and summable, then its values at natural numbers > 1
have norm strictly less than 1.
- Defined in
- Mathlib.NumberTheory.EulerProduct.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- Realstatement · cited by 25,697
- Norm.normstatement · cited by 5,413
- MonoidHomstatement and proof · cited by 3,629
- CompleteSpacestatement and proof · cited by 2,532
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Summablestatement and proof · cited by 778
- NormedDivisionRingstatement and proof · cited by 360
- Nat.pow_right_injectiveproof · cited by 23
- Summable.comp_injectiveproof · cited by 22
- summable_geometric_iff_norm_lt_oneproof · cited by 4
Cited by2
Results whose statement or proof uses this declaration.
- EulerProduct.exp_tsum_primes_log_eq_tsumproof · cited by 1
- EulerProduct.prod_filter_prime_geometric_eq_tsum_factoredNumbersproof · cited by 1