openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /ShortIntervals /MRTCharacterEulerShift.lean
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterEuler | |
| import OAI.NumberTheory.TwoPoint.Halasz.HalaszPrimeMass | |
| namespace OAI | |
| /-! Moving the finite character Euler product from Re(s)=1 to | |
| Re(s)=1+delta costs a bounded amount when delta=1/log X. All constants | |
| are independent of the character modulus and of the imaginary part. -/ | |
| namespace TwoPointCorrelations | |
| open Finset | |
| open scoped Classical ComplexConjugate | |
| lemma mrt_log_lipschitz_half {a b : ℝ} (ha : 1/2≤a) (hb : 1/2≤b) : | |
| |Real.log a-Real.log b|≤2*|a-b| := by | |
| have ha0 : 0<a := by linarith | |
| have hb0 : 0<b := by linarith | |
| have upper {x y : ℝ} (hx : 0<x) (hy : 1/2≤y) : | |
| Real.log x-Real.log y≤2*|x-y| := by | |
| have hy0 : 0<y := by linarith | |
| calc | |
| _ = Real.log (x/y) := (Real.log_div (ne_of_gt hx) (ne_of_gt hy0)).symm | |
| _ ≤ x/y-1 := Real.log_le_sub_one_of_pos (div_pos hx hy0) | |
| _ = (x-y)/y := by field_simp | |
| _ ≤ |x-y|/y := div_le_div_of_nonneg_right (le_abs_self _) (le_of_lt hy0) | |
| _ ≤ 2*|x-y| := (div_le_iff₀ hy0).mpr (by nlinarith [abs_nonneg (x-y)]) | |
| apply abs_le.mpr | |
| constructor | |
| · have hh := upper hb0 ha | |
| rw [abs_sub_comm b a] at hh | |
| linarith | |
| · exact upper ha0 hb | |
| lemma mrt_local_euler_log_lipschitz {z w : ℂ} | |
| (hz : ‖z‖≤1/2) (hw : ‖w‖≤1/2) : | |
| |Real.log ‖(1-z)⁻¹‖-Real.log ‖(1-w)⁻¹‖|≤2*‖z-w‖ := by | |
| have hza : 1/2≤‖(1:ℂ)-z‖ := by | |
| have hh := norm_sub_norm_le (1:ℂ) z | |
| rw [norm_one] at hh | |
| linarith | |
| have hwa : 1/2≤‖(1:ℂ)-w‖ := by | |
| have hh := norm_sub_norm_le (1:ℂ) w | |
| rw [norm_one] at hh | |
| linarith | |
| have hn : |‖(1:ℂ)-z‖-‖(1:ℂ)-w‖|≤‖z-w‖ := by | |
| have hh := abs_norm_sub_norm_le ((1:ℂ)-z) ((1:ℂ)-w) | |
| rw [show (1-z)-(1-w)=w-z by ring, norm_sub_rev w z] at hh | |
| exact hh | |
| calc | |
| _ = |Real.log ‖(1:ℂ)-z‖-Real.log ‖(1:ℂ)-w‖| := by | |
| rw [norm_inv,norm_inv,Real.log_inv,Real.log_inv,neg_sub_neg,abs_sub_comm] | |
| _ ≤ 2*|‖(1:ℂ)-z‖-‖(1:ℂ)-w‖| := mrt_log_lipschitz_half hza hwa | |
| _ ≤ 2*‖z-w‖ := mul_le_mul_of_nonneg_left hn (by norm_num) | |
| noncomputable def mrtCharacterShiftedEuler {q : ℕ} (χ : DirichletCharacter ℂ q) | |
| (t δ : ℝ) (S : Finset ℕ) : ℂ := | |
| ∏ p ∈ S, (1-(characterTwist χ t p/(p:ℂ))* | |
| (Real.exp (-δ*Real.log (p:ℝ)):ℂ))⁻¹ | |
| lemma mrt_character_euler_shift_local {q : ℕ} (χ : DirichletCharacter ℂ q) | |
| (t : ℝ) {δ : ℝ} (hδ : 0≤δ) {p : ℕ} (hp : p.Prime) : | |
| |Real.log ‖(1-characterTwist χ t p/(p:ℂ))⁻¹‖- | |
| Real.log ‖(1-(characterTwist χ t p/(p:ℂ))* | |
| (Real.exp (-δ*Real.log (p:ℝ)):ℂ))⁻¹‖| ≤ | |
| 2*δ*(Real.log (p:ℝ)/(p:ℝ)) := by | |
| let z : ℂ := characterTwist χ t p/(p:ℂ) | |
| let r : ℝ := Real.exp (-δ*Real.log (p:ℝ)) | |
| have hp0 : (0:ℝ)<p := by exact_mod_cast hp.pos | |
| have hp2 : (2:ℝ)≤p := by exact_mod_cast hp.two_le | |
| have hl : 0≤Real.log (p:ℝ) := Real.log_nonneg (by linarith) | |
| have hr0 : 0≤r := (Real.exp_pos _).le | |
| have hr1 : r≤1 := Real.exp_le_one_iff.mpr (by nlinarith) | |
| have hr : 1-r≤δ*Real.log (p:ℝ) := by | |
| have hh := Real.add_one_le_exp (-δ*Real.log (p:ℝ)) | |
| dsimp only [r] | |
| linarith | |
| have hz : ‖z‖≤1/(p:ℝ) := by | |
| dsimp only [z] | |
| rw [norm_div,Complex.norm_natCast] | |
| exact div_le_div_of_nonneg_right (characterTwist_norm_le_one χ t p) hp0.le | |
| have hz2 : ‖z‖≤1/2 := hz.trans (one_div_le_one_div_of_le (by norm_num) hp2) | |
| have hzr : ‖z*(r:ℂ)‖≤1/2 := by | |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr0] | |
| exact (mul_le_of_le_one_right (norm_nonneg z) hr1).trans hz2 | |
| have hd : ‖z-z*(r:ℂ)‖≤δ*(Real.log (p:ℝ)/(p:ℝ)) := by | |
| calc | |
| _ = ‖z‖*(1-r) := by | |
| rw [show z-z*(r:ℂ)=z*(1-(r:ℂ)) by ring, norm_mul, | |
| ← Complex.ofReal_one, ← Complex.ofReal_sub, | |
| Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hr1)] | |
| _ ≤ (1/(p:ℝ))*(δ*Real.log (p:ℝ)) := | |
| mul_le_mul hz hr (sub_nonneg.mpr hr1) (by positivity) | |
| _ = _ := by ring | |
| exact (mrt_local_euler_log_lipschitz hz2 hzr).trans | |
| (by simpa only [mul_assoc] using mul_le_mul_of_nonneg_left hd (by norm_num : (0:ℝ)≤2)) | |
| theorem mrt_character_euler_shift_bound {q : ℕ} (χ : DirichletCharacter ℂ q) | |
| (t : ℝ) {δ : ℝ} (hδ : 0≤δ) (X : ℕ) (S : Finset ℕ) | |
| (hS : S⊆primesUpTo X) : | |
| |Real.log ‖mrtCharacterPrimeEuler χ t S‖- | |
| Real.log ‖mrtCharacterShiftedEuler χ t δ S‖| ≤ | |
| 2*δ*(∑ p ∈ S, Real.log (p:ℝ)/(p:ℝ)) := by | |
| have hp (p : ℕ) (hh : p∈S) : p.Prime := (mem_filter.mp (hS hh)).2 | |
| have hz (p : ℕ) (hh : p∈S) : ‖characterTwist χ t p/(p:ℂ)‖≤1/2 := by | |
| rw [norm_div,Complex.norm_natCast] | |
| have h2 : (2:ℝ)≤p := by exact_mod_cast (hp p hh).two_le | |
| exact (div_le_div_of_nonneg_right (characterTwist_norm_le_one χ t p) | |
| (Nat.cast_nonneg p)).trans (one_div_le_one_div_of_le (by norm_num) h2) | |
| have hr (p : ℕ) (hh : p∈S) : | |
| 0≤Real.exp (-δ*Real.log (p:ℝ)) ∧ Real.exp (-δ*Real.log (p:ℝ))≤1 := by | |
| refine ⟨(Real.exp_pos _).le,Real.exp_le_one_iff.mpr ?_⟩ | |
| have h1 : (1:ℝ)≤p := by exact_mod_cast (hp p hh).one_le | |
| nlinarith [Real.log_nonneg h1] | |
| have hne {z : ℂ} (hh : ‖z‖≤1/2) : 1-z≠0 := by | |
| intro he | |
| have he' : z=1 := (sub_eq_zero.mp he).symm | |
| rw [he',norm_one] at hh | |
| norm_num at hh | |
| have hns (p : ℕ) (hh : p∈S) : | |
| 1-(characterTwist χ t p/(p:ℂ))*(Real.exp (-δ*Real.log (p:ℝ)):ℂ)≠0 := by | |
| apply hne | |
| rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hr p hh).1] | |
| exact (mul_le_of_le_one_right (norm_nonneg _) (hr p hh).2).trans (hz p hh) | |
| have hlog1 : Real.log ‖mrtCharacterPrimeEuler χ t S‖ = | |
| ∑ p ∈ S, Real.log ‖(1-characterTwist χ t p/(p:ℂ))⁻¹‖ := by | |
| rw [mrtCharacterPrimeEuler,norm_prod] | |
| exact Real.log_prod (fun p hh => norm_ne_zero_iff.mpr (inv_ne_zero (hne (hz p hh)))) | |
| have hlog2 : Real.log ‖mrtCharacterShiftedEuler χ t δ S‖ = | |
| ∑ p ∈ S, Real.log ‖(1-(characterTwist χ t p/(p:ℂ))* | |
| (Real.exp (-δ*Real.log (p:ℝ)):ℂ))⁻¹‖ := by | |
| rw [mrtCharacterShiftedEuler,norm_prod] | |
| exact Real.log_prod (fun p hh => norm_ne_zero_iff.mpr (inv_ne_zero (hns p hh))) | |
| rw [hlog1,hlog2,← sum_sub_distrib,Finset.mul_sum] | |
| exact (abs_sum_le_sum_abs _ _).trans | |
| (sum_le_sum (fun p hh => mrt_character_euler_shift_local χ t hδ (hp p hh))) | |
| theorem mrt_character_euler_logarithmic_shift {q : ℕ} | |
| (χ : DirichletCharacter ℂ q) (t : ℝ) {X : ℕ} | |
| (hX : 1≤Real.log (X:ℝ)) (S : Finset ℕ) (hS : S⊆primesUpTo X) : | |
| |Real.log ‖mrtCharacterPrimeEuler χ t S‖- | |
| Real.log ‖mrtCharacterShiftedEuler χ t (1/Real.log (X:ℝ)) S‖| ≤ | |
| 2+2*halaszMertensConstant := by | |
| have hl0 : 0<Real.log (X:ℝ) := by linarith | |
| have hX1 : (1:ℝ)≤X := by | |
| by_contra hh | |
| have hh' : (X:ℝ)<1 := lt_of_not_ge hh | |
| have hnatlt : X<1 := by exact_mod_cast hh' | |
| have hnat : X=0 := by omega | |
| norm_num [hnat] at hX | |
| have hs : (∑ p ∈ S, Real.log (p:ℝ)/(p:ℝ))≤ | |
| Real.log (X:ℝ)+halaszMertensConstant := by | |
| apply le_trans (sum_le_sum_of_subset_of_nonneg hS ?_) | |
| · simpa only [mrt_sievePrimesUpTo_nat] using halasz_prime_prefix_mass_le hX1 | |
| · intro p hp _ | |
| have hp1 : (1:ℝ)≤p := by exact_mod_cast (mem_filter.mp hp).2.one_le | |
| exact div_nonneg (Real.log_nonneg hp1) (Nat.cast_nonneg p) | |
| calc | |
| _ ≤ 2*(1/Real.log (X:ℝ))*(∑ p ∈ S, Real.log (p:ℝ)/(p:ℝ)) := | |
| mrt_character_euler_shift_bound χ t (by positivity) X S hS | |
| _ ≤ 2*(1/Real.log (X:ℝ))*(Real.log (X:ℝ)+halaszMertensConstant) := | |
| mul_le_mul_of_nonneg_left hs (by positivity) | |
| _ = 2+2*halaszMertensConstant/Real.log (X:ℝ) := by | |
| field_simp [ne_of_gt hl0] | |
| _ ≤ 2+2*halaszMertensConstant := by | |
| have hh := div_le_self | |
| (mul_nonneg (by norm_num : (0:ℝ)≤2) halaszMertensConstant_nonneg) hX | |
| linarith | |
| end TwoPointCorrelations | |
| end OAI | |