| import Mathlib.Analysis.SpecialFunctions.Pow.NNReal |
| import Mathlib.Topology.Algebra.InfiniteSum.ENNReal |
| import Mathlib.Topology.Algebra.Order.Field |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| The real-variable comparison controls the contribution of a nontrivial |
| zero after applying Stark (1974), Lemmas 3 and 8. The discriminant and |
| local-moment estimates relate the arithmetic counting scales. |
| -/ |
|
|
| open Filter |
| open scoped Topology ENNReal |
|
|
| namespace DukePrimeDegree |
|
|
| /-- Uniform comparison of a zero's contribution to the logarithmic |
| derivative, once its rescaled distance from 1 is bounded below. |
| The constant is slightly larger than the manuscript's, which is harmless. -/ |
| theorem zero_contribution_comparison |
| {X Y U c : ℝ} (hX : 0 ≤ X) (hU : 0 < U) (hU4 : U ≤ 4) |
| (hc : 0 < c) (hgap : c ^ 2 ≤ X ^ 2 + Y ^ 2) : |
| (X + U) / ((X + U) ^ 2 + Y ^ 2) ≤ |
| (8 + 8 / c ^ 2) * ((X + 1) / ((X + 1) ^ 2 + Y ^ 2)) := by |
| have hc2 : 0 < c ^ 2 := sq_pos_of_pos hc |
| have hd : 0 < (X + U) ^ 2 + Y ^ 2 := by positivity |
| have he : 0 < (X + 1) ^ 2 + Y ^ 2 := by positivity |
| have hsmall : X ^ 2 + Y ^ 2 ≤ (X + U) ^ 2 + Y ^ 2 := by nlinarith |
| have hratio : (X + 1) ^ 2 + Y ^ 2 ≤ |
| (2 + 2 / c ^ 2) * ((X + U) ^ 2 + Y ^ 2) := by |
| have hbound : c ^ 2 ≤ (X + U) ^ 2 + Y ^ 2 := hgap.trans hsmall |
| have hdiv : 2 ≤ (2 / c ^ 2) * ((X + U) ^ 2 + Y ^ 2) := by |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ hc2).mpr |
| nlinarith |
| have hpoly : (X + 1) ^ 2 + Y ^ 2 ≤ |
| 2 * ((X + U) ^ 2 + Y ^ 2) + 2 := by nlinarith [sq_nonneg (X - 1), sq_nonneg Y] |
| nlinarith |
| have hnum : X + U ≤ 4 * (X + 1) := by linarith |
| apply (div_le_iff₀ hd).mpr |
| rw [mul_assoc, div_mul_eq_mul_div, ← mul_div_assoc, le_div_iff₀ he] |
| have hprod := mul_le_mul hnum hratio (le_of_lt he) (by positivity : 0 ≤ 4 * (X + 1)) |
| convert hprod using 1; ring |
|
|
| /-- The packet discriminant bound suffices to transfer divergence |
| from the multiplier-order discriminant to the counting scale. -/ |
| theorem scale_tendsto_atTop {D Q : ℕ → ℝ} {k : ℕ} |
| (hQ : ∀ i, 0 ≤ Q i) (hbound : ∀ i, D i ≤ Q i ^ k) |
| (hD : Tendsto D atTop atTop) : Tendsto Q atTop atTop := by |
| apply tendsto_atTop.2 |
| intro b |
| filter_upwards [(tendsto_atTop.1 hD) ((max b 0) ^ k + 1)] with i hi |
| have hlt : max b 0 < Q i := by |
| by_contra h |
| have hp := pow_le_pow_left₀ (hQ i) (le_of_not_gt h) k |
| linarith [hbound i] |
| exact (le_max_left b 0).trans hlt.le |
|
|
| /-- Markov's estimate for arbitrary (possibly correlated) local shells. |
| All sums are extended nonnegative sums, so no convergence hypothesis is |
| needed. This is the tail extraction used in `lem:local-moment`. -/ |
| theorem weighted_tail_le_moment {α : Type*} (mass size : α → ℝ≥0∞) |
| (R : ℝ≥0∞) (s : ℝ) (hs : 0 ≤ s) (hR : R ≠ 0) (hRtop : R ≠ ∞) : |
| (∑' i, if R < size i then mass i else 0) ≤ |
| (R ^ s)⁻¹ * ∑' i, mass i * size i ^ s := by |
| have hpow : R ^ s ≠ 0 := by |
| exact (ENNReal.rpow_pos (pos_iff_ne_zero.mpr hR) hRtop).ne' |
| calc |
| (∑' i, if R < size i then mass i else 0) |
| ≤ ∑' i, (R ^ s)⁻¹ * (mass i * size i ^ s) := by |
| apply ENNReal.tsum_le_tsum |
| intro i |
| split_ifs with hi |
| · calc |
| mass i = (R ^ s)⁻¹ * (mass i * R ^ s) := by |
| rw [mul_left_comm, ENNReal.inv_mul_cancel hpow] |
| · simp |
| · exact ENNReal.rpow_ne_top_of_nonneg hs hRtop |
| _ ≤ _ := mul_le_mul_right |
| (mul_le_mul_right (ENNReal.rpow_le_rpow hi.le hs) (mass i)) _ |
| · exact zero_le |
| _ = _ := ENNReal.tsum_mul_left |
|
|
| /-- Absorb the harmless `q^(s/16)` loss in the moment bound into the |
| cutoff `Q^(1/8)`. This is the exponent calculation giving `Q^(-s/16)` |
| in the local-weight tail estimate, using only `q ≤ Q`. -/ |
| theorem tail_exponent_absorption {q Q s : ℝ} (hq : 0 ≤ q) (hQ : 0 < Q) |
| (hqQ : q ≤ Q) (hs : 0 ≤ s) : |
| Q ^ (-s / 8) * q ^ (s / 16) ≤ Q ^ (-s / 16) := by |
| calc |
| Q ^ (-s / 8) * q ^ (s / 16) ≤ Q ^ (-s / 8) * Q ^ (s / 16) := |
| mul_le_mul_of_nonneg_left (Real.rpow_le_rpow hq hqQ (by positivity)) |
| (Real.rpow_nonneg hQ.le _) |
| _ = Q ^ (-s / 16) := by |
| rw [← Real.rpow_add hQ] |
| congr 1 |
| ring |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
|
|
