Theorems · Theorem · number theory

NumberField.Ideal.tendsto_norm_le_div_atTop

∀ (K : Type u_1) [inst : Field K] [inst_1 : NumberField K],
  Filter.Tendsto (fun s => ↑(Nat.card { I // ↑(Ideal.absNorm I) ≤ s }) / s) Filter.atTop
    (nhds
      (2 ^ NumberField.InfinitePlace.nrRealPlaces K * (2 * Real.pi) ^ NumberField.InfinitePlace.nrComplexPlaces K *
            NumberField.Units.regulator K *
          ↑(NumberField.classNumber K) /
        (↑(NumberField.Units.torsionOrder K) * √|↑(NumberField.discr K)|)))

The limit of the number of integral ideals of norm ≤ s divided by s when s → +∞.

Defined in: Mathlib.NumberTheory.NumberField.Ideal.Asymptotics
Cited by: 0 results in Mathlib
Foundations: Depth 348 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Field NumberField

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DFunLike.coestatement and proof · cited by 62,936
Realstatement and proof · cited by 25,697
Fieldstatement and proof · cited by 7,404
nhdsstatement and proof · cited by 5,554
Idealstatement and proof · cited by 4,748
Filter.Tendstostatement and proof · cited by 3,814
add_zeroproof · cited by 2,707
Nat.cast_oneproof · cited by 2,501
Filter.atTopstatement and proof · cited by 2,405
Filter.EventuallyEqproof · cited by 1,912
absstatement and proof · cited by 1,814
Real.pistatement and proof · cited by 1,774

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