| import OAI.NumberTheory.Ostmann.ZeroDensity.ProgressionErrorRates |
|
|
| namespace OAI |
|
|
| /-! # Giant and bulk progression rows used in the arithmetic comparison -/ |
|
|
| namespace Ostmann |
|
|
| open Filter |
|
|
| inductive ArithmeticProgressionRow |
| | giant |
| | bulk |
| deriving DecidableEq |
|
|
| namespace ArithmeticProgressionRow |
|
|
| noncomputable def leftExponent : ArithmeticProgressionRow → ℝ |
| | giant => 49 / 1000 |
| | bulk => 39 / 10000 |
|
|
| noncomputable def modulusExponent : ArithmeticProgressionRow → ℝ |
| | giant => 12 / 1000 |
| | bulk => 12 / 10000 |
|
|
| noncomputable def errorExponent : ArithmeticProgressionRow → ℝ |
| | giant => 18 / 1000 |
| | bulk => 16 / 10000 |
|
|
| theorem gaps (r : ArithmeticProgressionRow) : |
| 0 < r.leftExponent ∧ 0 < r.modulusExponent ∧ |
| r.errorExponent < r.leftExponent / 2 ∧ |
| r.errorExponent < r.leftExponent - r.modulusExponent ∧ |
| 0 < r.leftExponent - r.modulusExponent := by |
| cases r <;> norm_num [leftExponent, modulusExponent, errorExponent] |
|
|
| end ArithmeticProgressionRow |
|
|
| / |
| exceptional character for all moduli in the row. The logarithmic interval |
| estimate is proved from the published theta and Page inputs. -/ |
| theorem PublishedProgressionInput.row_log_interval (P : PublishedProgressionInput) |
| (r : ArithmeticProgressionRow) : |
| ∀ᶠ L : ℝ in atTop, ∀ (q Q a : ℕ), 2 ≤ Q → 1 ≤ q → q ≤ Q → a.Coprime q → |
| Real.log (Q : ℝ) ≤ Real.exp (r.modulusExponent * L) → |
| ∀ s t : ℝ, 1 ≤ s → Real.exp (r.leftExponent * L) ≤ s → s ≤ t → t ≤ s + 1 → |
| |reciprocalPrimeInterval q a (Real.exp s) (Real.exp t) - |
| ∫ y in Set.Ioc s t, primeLogDensity (Nat.totient q) |
| (pageCoefficient (pageAtModulus q (selectedPageZero P Q)) a) |
| (pageBeta (pageAtModulus q (selectedPageZero P Q))) y| ≤ |
| 2 * Real.exp (-Real.exp (r.errorExponent * L)) := by |
| obtain ⟨ha, hμ, hδa, hδμ, haμ⟩ := r.gaps |
| filter_upwards [P.log_interval_rate r.leftExponent r.modulusExponent r.errorExponent |
| ha hδa hδμ haμ, |
| eventually_ge_atTop (Real.log 4 / r.modulusExponent)] with L hL hLlarge |
| intro q Q a hQ hq hqQ hcop hlog |
| apply hL q Q a hQ hq hqQ hcop |
| have hQ0 : (Q : ℝ) ≠ 0 := by exact_mod_cast (by omega : Q ≠ 0) |
| rw [Real.log_mul (by norm_num : (4 : ℝ) ≠ 0) hQ0] |
| have hm : Real.log 4 ≤ r.modulusExponent * L := by |
| have h := (div_le_iff₀ hμ).mp hLlarge |
| nlinarith |
| have he := Real.add_one_le_exp (r.modulusExponent * L) |
| linarith |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
