| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterContinuation |
| import Mathlib.NumberTheory.AbelSummation |
|
|
| namespace OAI |
|
|
| /-! Ordered harmonic sums for a nonprincipal character converge to the |
| existing analytic L-function at one. No absolute convergence at one is used. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset Filter MeasureTheory Asymptotics |
| open scoped Classical Topology |
|
|
| variable {q : ℕ} [NeZero q] |
|
|
| lemma mrt_character_zero (χ : DirichletCharacter ℂ q) (hχ : χ ≠ 1) : |
| χ (0 : ZMod q) = 0 := by |
| have hq : q ≠ 1 := by |
| intro hq |
| subst q |
| apply hχ |
| ext n |
| have hn : n = (1 : ZMod 1) := Subsingleton.elim _ _ |
| simp [hn] |
| let : Nontrivial (ZMod q) := ZMod.nontrivial_iff.mpr hq |
| exact MulChar.map_zero χ |
|
|
| lemma mrt_character_Icc_zero_eq (χ : DirichletCharacter ℂ q) (hχ : χ ≠ 1) |
| (N : ℕ) : |
| (∑ n ∈ Icc 0 N, χ (n : ZMod q)) = ∑ n ∈ Icc 1 N, χ (n : ZMod q) := by |
| rw [← insert_Icc_add_one_left_eq_Icc N.zero_le, sum_insert (by simp), |
| Nat.cast_zero, mrt_character_zero χ hχ, zero_add, zero_add] |
|
|
| theorem mrt_character_harmonic_tendsto (χ : DirichletCharacter ℂ q) (hχ : χ ≠ 1) : |
| Tendsto (fun N : ℕ => ∑ n ∈ Icc 1 N, χ (n : ZMod q) / (n : ℂ)) |
| atTop (𝓝 (DirichletCharacter.LFunction χ 1)) := by |
| let c : ℕ → ℂ := fun n => χ (n : ZMod q) |
| let f : ℝ → ℂ := fun t => (t : ℂ) ^ (-1 : ℂ) |
| have hc : c 0 = 0 := by simpa only [c, Nat.cast_zero] using mrt_character_zero χ hχ |
| have hsum (N : ℕ) : (∑ n ∈ Icc 0 N, c n) = ∑ n ∈ Icc 1 N, χ (n : ZMod q) := |
| mrt_character_Icc_zero_eq χ hχ N |
| have hdiff (t : ℝ) (ht : t ∈ Set.Ici (1 : ℝ)) : DifferentiableAt ℝ f t := |
| differentiableAt_id.ofReal_cpow_const (zero_lt_one.trans_le ht).ne' (by norm_num) |
| have hint : LocallyIntegrableOn (deriv f) (Set.Ici (1 : ℝ)) := |
| (Iff.mpr integrableOn_Ici_iff_integrableOn_Ioi |
| (integrableOn_Ioi_deriv_ofReal_cpow zero_lt_one (by norm_num : (-1 : ℂ).re < 0))).locallyIntegrableOn |
| have hlim : Tendsto (fun N : ℕ => f N * ∑ n ∈ Icc 0 N, c n) atTop (𝓝 0) := by |
| apply squeeze_zero_norm (a := fun N : ℕ => (q : ℝ) * (N : ℝ)⁻¹) |
| · intro N |
| dsimp only [f] |
| rw [norm_mul, hsum, Complex.cpow_neg_one, norm_inv, Complex.norm_real, |
| Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg N)] |
| calc |
| _ ≤ (N : ℝ)⁻¹ * q := mul_le_mul_of_nonneg_left |
| (mrt_character_Icc_norm χ hχ N) (inv_nonneg.mpr (Nat.cast_nonneg N)) |
| _ = _ := mul_comm _ _ |
| · simpa using Tendsto.const_mul (q : ℝ) |
| (tendsto_inv_atTop_zero.comp tendsto_natCast_atTop_atTop) |
| have hO : (fun N : ℕ => ∑ n ∈ Icc 0 N, c n) =O[atTop] |
| (fun N : ℕ => (N : ℝ) ^ (0 : ℝ)) := by |
| simpa only [hsum] using mrt_character_sum_bigO χ hχ |
| have hderiv : (fun t : ℝ => deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n) =O[atTop] |
| (fun t : ℝ => t ^ (-2 : ℝ)) := by |
| refine IsBigO.mul_atTop_rpow_of_isBigO_rpow (-2) 0 (-2) ?_ ?_ (by norm_num) |
| · have hh := isBigO_deriv_ofReal_cpow_const_atTop (-1 : ℂ) |
| norm_num only [Complex.neg_re, Complex.one_re] at hh |
| simpa only [f] using! hh |
| · simpa only [Real.rpow_zero, Function.comp_apply] using! |
| hO.comp_tendsto tendsto_nat_floor_atTop |
| have ht := tendsto_sum_mul_atTop_nhds_one_sub_integral₀ c hc hdiff hint hlim hderiv |
| (integrableAtFilter_rpow_atTop_iff.mpr (by norm_num : (-2 : ℝ) < -1)) |
| have he : (0 : ℂ) - ∫ t in Set.Ioi (1 : ℝ), deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n = |
| DirichletCharacter.LFunction χ 1 := by |
| rw [zero_sub, ← integral_neg, |
| ← mrtCharacterAbelIntegral_eq_LFunction_of_pos χ hχ (by norm_num : 0 < (1 : ℂ).re)] |
| simp only [mrtCharacterAbelIntegral, one_mul] |
| apply setIntegral_congr_fun measurableSet_Ioi |
| intro t ht |
| change -(deriv f t * ∑ n ∈ Icc 0 ⌊t⌋₊, c n) = mrtCharacterAbelKernel χ 1 t |
| rw [show deriv f t = (-1 : ℂ) * (t : ℂ) ^ (-2 : ℂ) by |
| have hh := Complex.deriv_ofReal_cpow_const |
| (zero_lt_one.trans ht).ne' (by norm_num : (-1 : ℂ) ≠ 0) |
| norm_num only at hh |
| exact hh, hsum] |
| simp only [mrtCharacterAbelKernel, mrtCharacterSummatory] |
| norm_num |
| ring |
| rw [he] at ht |
| convert ht using 1 |
| funext N |
| have hz : (f 0) * c 0 = 0 := by rw [hc, mul_zero] |
| rw [← insert_Icc_add_one_left_eq_Icc N.zero_le, sum_insert (by simp), Nat.cast_zero, hz, zero_add, |
| zero_add] |
| apply sum_congr rfl |
| intro n _ |
| simp [f, c, Complex.cpow_neg_one, div_eq_mul_inv, mul_comm] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
