| import OAI.NumberTheory.TwoPointCorrelations.HalaszPrimeTruncation |
| import OAI.NumberTheory.TwoPointCorrelations.HalaszInnerApproximation |
|
|
| namespace OAI |
|
|
| /-! Apply the prime convolution a second time, retaining the varying |
| denominator log(N/p). This is the adaptive triple sum in the short proof of |
| Halasz's theorem. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| noncomputable def halaszDoubleConvolution (f : ℕ → ℂ) (N : ℕ) (L : ℝ) : ℂ := |
| ∑ p ∈ mrtPrimeBand L ((N : ℝ) / 2), |
| (Real.log (p : ℝ) : ℂ) * f p * |
| (halaszPrimeConvolution f (N / p) / (Real.log ((N : ℝ) / p) : ℂ)) |
|
|
| lemma halasz_divisor_cutoff {N p : ℕ} (hp : 0 < p) (hpN : 2 * p ≤ N) : |
| 1 ≤ N / p ∧ ((N / p : ℕ) : ℝ) ≤ (N : ℝ) / p ∧ |
| (N : ℝ) / p ≤ 2 * ((N / p : ℕ) : ℝ) := by |
| have hp0 : (0 : ℝ) < p := by exact_mod_cast hp |
| have hM : 1 ≤ N / p := (Nat.le_div_iff_mul_le hp).mpr (by omega) |
| refine ⟨hM, ?_, ?_⟩ |
| · apply (le_div_iff₀ hp0).mpr |
| exact_mod_cast Nat.div_mul_le_self N p |
| · have hmod : ((N % p : ℕ) : ℝ) < p := by exact_mod_cast Nat.mod_lt N hp |
| have hid : ((N % p : ℕ) : ℝ) + (p : ℝ) * ((N / p : ℕ) : ℝ) = N := by |
| exact_mod_cast Nat.mod_add_div N p |
| have hM1 : (1 : ℝ) ≤ ((N / p : ℕ) : ℝ) := by exact_mod_cast hM |
| apply (div_le_iff₀ hp0).mpr |
| nlinarith |
|
|
| theorem halasz_double_convolution_error (f : ℕ → ℂ) (hf : OneBounded f) |
| (hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n) |
| (N : ℕ) {L : ℝ} (hL : 1 ≤ L) : |
| ‖halaszPrimeBandConvolution f N L ((N : ℝ) / 2) - |
| halaszDoubleConvolution f N L‖ ≤ |
| halaszInnerErrorConstant * (N : ℝ) * |
| ∑ p ∈ mrtPrimeBand L ((N : ℝ) / 2), |
| Real.log (p : ℝ) / ((p : ℝ) * Real.log ((N : ℝ) / p)) := by |
| have hN : (0 : ℝ) ≤ N := Nat.cast_nonneg _ |
| have hL0 : 0 ≤ L := by linarith |
| unfold halaszPrimeBandConvolution halaszDoubleConvolution halaszPrimeTerm |
| rw [← sum_sub_distrib, mul_sum] |
| apply (norm_sum_le _ _).trans |
| apply sum_le_sum |
| intro p hp |
| have hprime := mrtPrimeBand_prime hp |
| have hb := mrtPrimeBand_bounds hL0 (by positivity : (0 : ℝ) ≤ (N : ℝ) / 2) hp |
| have hpN : 2 * p ≤ N := by exact_mod_cast (show 2 * (p : ℝ) ≤ N by linarith) |
| obtain ⟨hM, hMx, hxM⟩ := halasz_divisor_cutoff hprime.pos hpN |
| have hp0 : (0 : ℝ) < p := by exact_mod_cast hprime.pos |
| have hx : 1 < (N : ℝ) / p := by |
| have hh : (2 : ℝ) ≤ (N : ℝ) / p := (le_div_iff₀ hp0).mpr (by linarith) |
| linarith |
| have hlog : 0 ≤ Real.log (p : ℝ) := Real.log_nonneg (by exact_mod_cast hprime.one_le) |
| rw [← mul_sub, norm_mul, norm_mul, Complex.norm_real, Real.norm_eq_abs, |
| abs_of_nonneg hlog] |
| calc |
| _ ≤ Real.log (p : ℝ) * 1 * |
| (halaszInnerErrorConstant * ((N : ℝ) / p) / Real.log ((N : ℝ) / p)) := by |
| gcongr |
| · exact hf p hprime.pos |
| · exact halasz_real_inner_approximation f hf hmul (N / p) hM hx hMx hxM |
| _ = _ := by ring |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
