| import Mathlib.Analysis.Complex.Trigonometric |
| import Mathlib.NumberTheory.DirichletCharacter.Bounds |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! |
| # The pretentious distance used in ordinary two-point correlations |
|
|
| These definitions use the prime cutoff and the twist-height cutoff at the same |
| integer scale. The finite-prime perturbation estimate below is Lemma |
| `lem:finite-prime-stability` of *Ordinary two-point correlations of |
| multiplicative functions* (24 September 2026). It is elementary and does not |
| assume any analytic input. |
| -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open scoped ComplexConjugate |
| open Filter Finset |
|
|
| / |
| def primesUpTo (N : ℕ) : Finset ℕ := (Finset.range (N + 1)).filter Nat.Prime |
|
|
| / |
| noncomputable def squaredDistance (f g : ℕ → ℂ) (N : ℕ) : ℝ := |
| ∑ p ∈ primesUpTo N, (1 - (f p * conj (g p)).re) / (p : ℝ) |
|
|
| / |
| noncomputable def characterTwist {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t : ℝ) (n : ℕ) : ℂ := |
| χ (n : ZMod q) * Complex.exp (((t * Real.log (n : ℝ) : ℝ) : ℂ) * Complex.I) |
|
|
| / |
|
|
| The quantifier over all real lower bounds expresses divergence of the squared |
| distance uniformly over `|t| ≤ N`. As the distance is nonnegative, this is |
| exactly the divergence of the distance appearing in the manuscript. |
| -/ |
| def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := |
| ∀ (q : ℕ), 0 < q → ∀ (χ : DirichletCharacter ℂ q) (K : ℝ), |
| ∀ᶠ N : ℕ in atTop, ∀ t : ℝ, |t| ≤ (N : ℝ) → |
| K ≤ squaredDistance f (characterTwist χ t) N |
|
|
| lemma characterTwist_norm_le_one {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t : ℝ) (n : ℕ) : ‖characterTwist χ t n‖ ≤ 1 := by |
| simpa only [characterTwist, norm_mul, Complex.norm_exp_ofReal_mul_I, mul_one] using |
| χ.norm_le_one (n : ZMod q) |
|
|
| lemma squaredDistance_nonneg (f g : ℕ → ℂ) (N : ℕ) |
| (hf : ∀ n, ‖f n‖ ≤ 1) (hg : ∀ n, ‖g n‖ ≤ 1) : |
| 0 ≤ squaredDistance f g N := by |
| apply Finset.sum_nonneg |
| intro p hp |
| apply div_nonneg |
| · have hnorm : ‖f p * conj (g p)‖ ≤ 1 := by |
| calc |
| ‖f p * conj (g p)‖ = ‖f p‖ * ‖g p‖ := by simp |
| _ ≤ 1 * 1 := mul_le_mul (hf p) (hg p) (norm_nonneg _) zero_le_one |
| _ = 1 := by norm_num |
| exact sub_nonneg.mpr ((Complex.re_le_norm _).trans hnorm) |
| · positivity |
|
|
| lemma squaredDistance_conj (f g : ℕ → ℂ) (N : ℕ) : |
| squaredDistance (fun n => conj (f n)) (fun n => conj (g n)) N = |
| squaredDistance f g N := by |
| unfold squaredDistance |
| apply Finset.sum_congr rfl |
| intro p hp |
| congr 2 |
| rw [← map_mul] |
| simp |
|
|
| / |
| lemma characterTwist_conj {q : ℕ} (χ : DirichletCharacter ℂ q) |
| (t : ℝ) (n : ℕ) : |
| characterTwist (χ.ringHomComp (starRingEnd ℂ)) (-t) n = |
| conj (characterTwist χ t n) := by |
| simp only [characterTwist, MulChar.ringHomComp_apply, map_mul, |
| ← Complex.exp_conj] |
| congr 2 |
| simp only [neg_mul, Complex.ofReal_neg, Complex.conj_ofReal, Complex.conj_I, mul_neg] |
|
|
| / |
| lemma UniformlyNonpretentious.conj {f : ℕ → ℂ} |
| (hf : UniformlyNonpretentious f) : |
| UniformlyNonpretentious (fun n => conj (f n)) := by |
| intro q hq χ K |
| filter_upwards [hf q hq (χ.ringHomComp (starRingEnd ℂ)) K] with N hN |
| intro t ht |
| have h := hN (-t) (by simpa using ht) |
| have heq : characterTwist (χ.ringHomComp (starRingEnd ℂ)) (-t) = |
| fun n => conj (characterTwist χ t n) := by |
| funext n |
| exact characterTwist_conj χ t n |
| rw [heq] at h |
| simpa only [← squaredDistance_conj f (fun n => conj (characterTwist χ t n)) N, |
| Complex.conj_conj] using h |
|
|
| / |
| pretentious distance by at most twice their reciprocal sum. -/ |
| lemma squaredDistance_finite_prime_change (f b g : ℕ → ℂ) (N : ℕ) |
| (P : Finset ℕ) (hf : ∀ n, 0 < n → ‖f n‖ ≤ 1) (hb : ∀ n, 0 < n → ‖b n‖ ≤ 1) |
| (hg : ∀ n, 0 < n → ‖g n‖ ≤ 1) |
| (heq : ∀ p, Nat.Prime p → p ∉ P → b p = f p) : |
| |squaredDistance b g N - squaredDistance f g N| ≤ |
| 2 * ∑ p ∈ P, (1 : ℝ) / (p : ℝ) := by |
| let S := primesUpTo N |
| have hterm (p : ℕ) (hp : 0 < p) : |
| |(1 - (b p * conj (g p)).re) / (p : ℝ) - |
| (1 - (f p * conj (g p)).re) / (p : ℝ)| ≤ 2 / (p : ℝ) := by |
| have hdiff : ‖f p - b p‖ ≤ 2 := by |
| exact (norm_sub_le _ _).trans (by linarith [hf p hp, hb p hp]) |
| have hprod : ‖(f p - b p) * conj (g p)‖ ≤ 2 := by |
| calc |
| ‖(f p - b p) * conj (g p)‖ = ‖f p - b p‖ * ‖g p‖ := by simp |
| _ ≤ 2 * 1 := mul_le_mul hdiff (hg p hp) (norm_nonneg _) (by norm_num) |
| _ = 2 := by norm_num |
| have hre : |(f p * conj (g p)).re - (b p * conj (g p)).re| ≤ 2 := by |
| calc |
| |(f p * conj (g p)).re - (b p * conj (g p)).re| = |
| |((f p - b p) * conj (g p)).re| := by rw [sub_mul, Complex.sub_re] |
| _ ≤ ‖(f p - b p) * conj (g p)‖ := Complex.abs_re_le_norm _ |
| _ ≤ 2 := hprod |
| rw [← sub_div, show (1 - (b p * conj (g p)).re) - |
| (1 - (f p * conj (g p)).re) = |
| (f p * conj (g p)).re - (b p * conj (g p)).re by ring, |
| abs_div, show |(p : ℝ)| = (p : ℝ) from abs_of_nonneg (Nat.cast_nonneg p)] |
| exact div_le_div_of_nonneg_right hre (Nat.cast_nonneg p) |
| have htermP (p : ℕ) (hp : p ∈ S) : |
| |(1 - (b p * conj (g p)).re) / (p : ℝ) - |
| (1 - (f p * conj (g p)).re) / (p : ℝ)| ≤ |
| if p ∈ P then 2 / (p : ℝ) else 0 := by |
| split_ifs with hP |
| · exact hterm p ((Finset.mem_filter.mp hp).2.pos) |
| · have hprime : Nat.Prime p := (Finset.mem_filter.mp hp).2 |
| rw [heq p hprime hP, sub_self, abs_zero] |
| calc |
| |squaredDistance b g N - squaredDistance f g N| = |
| |∑ p ∈ S, ((1 - (b p * conj (g p)).re) / (p : ℝ) - |
| (1 - (f p * conj (g p)).re) / (p : ℝ))| := by |
| rw [Finset.sum_sub_distrib] |
| rfl |
| _ ≤ ∑ p ∈ S, |(1 - (b p * conj (g p)).re) / (p : ℝ) - |
| (1 - (f p * conj (g p)).re) / (p : ℝ)| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ p ∈ S, if p ∈ P then 2 / (p : ℝ) else 0 := Finset.sum_le_sum htermP |
| _ = ∑ p ∈ S.filter (fun p => p ∈ P), 2 / (p : ℝ) := by rw [Finset.sum_filter] |
| _ ≤ ∑ p ∈ P, 2 / (p : ℝ) := by |
| apply Finset.sum_le_sum_of_subset_of_nonneg |
| · intro p hp |
| exact (Finset.mem_filter.mp hp).2 |
| · intro p hp hnot |
| positivity |
| _ = 2 * ∑ p ∈ P, (1 : ℝ) / (p : ℝ) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro p hp |
| ring |
|
|
| / |
| lemma UniformlyNonpretentious.finite_prime_change {f b : ℕ → ℂ} |
| (hfnp : UniformlyNonpretentious f) (P : Finset ℕ) |
| (hf : ∀ n, 0 < n → ‖f n‖ ≤ 1) (hb : ∀ n, 0 < n → ‖b n‖ ≤ 1) |
| (heq : ∀ p, Nat.Prime p → p ∉ P → b p = f p) : |
| UniformlyNonpretentious b := by |
| intro q hq χ K |
| filter_upwards [hfnp q hq χ (K + 2 * ∑ p ∈ P, (1 : ℝ) / (p : ℝ))] with N hN |
| intro t ht |
| have hdist := squaredDistance_finite_prime_change f b (characterTwist χ t) N |
| P hf hb (fun n _ => characterTwist_norm_le_one χ t n) heq |
| have hlow := hN t ht |
| have := (abs_le.mp hdist).1 |
| linarith |
|
|
| / |
| and for every bounded sequence whose prime values are changed only in `P`. |
| This is the uniformity needed when taking the short-interval theorem at a fixed |
| auxiliary graph scale. -/ |
| lemma UniformlyNonpretentious.finite_family {ι : Type*} [Finite ι] |
| {f : ℕ → ℂ} (hfnp : UniformlyNonpretentious f) (hf : ∀ n, 0 < n → ‖f n‖ ≤ 1) |
| (P : Finset ℕ) (q : ι → ℕ) (hq : ∀ i, 0 < q i) |
| (χ : ∀ i, DirichletCharacter ℂ (q i)) (K : ℝ) : |
| ∀ᶠ N : ℕ in atTop, ∀ i (b : ℕ → ℂ), |
| (∀ n, 0 < n → ‖b n‖ ≤ 1) → |
| (∀ p, Nat.Prime p → p ∉ P → b p = f p) → |
| ∀ t : ℝ, |t| ≤ (N : ℝ) → K ≤ squaredDistance b (characterTwist (χ i) t) N := by |
| have hfamily : ∀ᶠ N : ℕ in atTop, ∀ i, |
| ∀ t : ℝ, |t| ≤ (N : ℝ) → |
| K + 2 * ∑ p ∈ P, (1 : ℝ) / (p : ℝ) ≤ |
| squaredDistance f (characterTwist (χ i) t) N := by |
| rw [Filter.eventually_all] |
| intro i |
| exact hfnp (q i) (hq i) (χ i) _ |
| filter_upwards [hfamily] with N hN |
| intro i b hb heq t ht |
| have hdist := squaredDistance_finite_prime_change f b (characterTwist (χ i) t) N |
| P hf hb (fun n _ => characterTwist_norm_le_one (χ i) t n) heq |
| have hlow := hN i t ht |
| have := (abs_le.mp hdist).1 |
| linarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
