Theorems · Theorem · number theory
ArithmeticFunction.vonMangoldt.not_summable_residueClass_prime_div
∀ {q : ℕ} [NeZero q] {a : ZMod q},
IsUnit a → ¬Summable fun n => (if Nat.Prime n then ArithmeticFunction.vonMangoldt.residueClass a n else 0) / ↑nThe function n ↦ Λ n / n restricted to primes in an invertible residue class
is not summable. This then implies that there must be infinitely many such primes.
- Defined in
- Mathlib.NumberTheory.LSeries.PrimesInAP
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 321 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- NeZero
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites54
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- TopologicalSpaceproof · cited by 24,529
- AddCommMonoidproof · cited by 12,281
- mul_oneproof · cited by 3,885
- LE.le.transproof · cited by 3,151
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- LT.lt.leproof · cited by 2,189
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- Nat.Primestatement and proof · cited by 2,059
- Nat.cast_zeroproof · cited by 1,870
Cited by1
Results whose statement or proof uses this declaration.
- Nat.infinite_setOfPred_prime_and_eq_modproof · cited by 2