| import OAI.NumberTheory.PrimeGaps.ConductorBounds | |
| namespace OAI | |
| namespace LargePrimeGaps | |
| open Filter | |
| open Set Filter MeasureTheory | |
| open scoped Topology ContDiff | |
| open Asymptotics | |
| open Asymptotics | |
| open Asymptotics | |
| open scoped Classical | |
| open scoped ContDiff | |
| open Topology | |
| open scoped Convolution ContDiff Pointwise | |
| open scoped ComplexConjugate | |
| open Filter Topology | |
| theorem exists_vaughan_log_parameters {A : ℝ} (hA : 0 < A) {X : ℕ} | |
| (hX : 16 ≤ X) (hL : 2 ≤ Real.log (X:ℝ)) | |
| (hpower : (Real.log (X:ℝ))^(A+10) ≤ (X:ℝ)^(1/32:ℝ)) : | |
| ∃ T k r n : ℕ, | |
| X+1 ≤ 2^k ∧ | |
| 2*(X:ℝ)^(1/2:ℝ)*(Real.log (X:ℝ))^(-A-10) ≤ ((2^(r+n):ℕ):ℝ) ∧ | |
| 2^(r+n) ≤ X ∧ | |
| ((2^(r+n):ℕ):ℝ) ≤ 4*(X:ℝ)^(1/2:ℝ)*(Real.log (X:ℝ))^(-A-10) ∧ | |
| (Real.log (X:ℝ))^(A+10) ≤ ((2^r:ℕ):ℝ) ∧ | |
| ((2^r:ℕ):ℝ) ≤ 2*(Real.log (X:ℝ))^(A+10) ∧ | |
| (n:ℝ) ≤ 6*Real.log (X:ℝ) ∧ | |
| (k:ℝ) ≤ 6*Real.log (X:ℝ) ∧ | |
| (T:ℝ) ≤ (X:ℝ)^(1/16:ℝ) ∧ | |
| (X:ℝ)^(1/32:ℝ) ≤ Real.sqrt (T+1:ℕ) := by | |
| let x : ℝ := X | |
| let L : ℝ := Real.log x | |
| let B : ℝ := A+10 | |
| let y : ℝ := 2*x^(1/2:ℝ)*L^(-B) | |
| let T : ℕ := ⌊x^(1/16:ℝ)⌋₊ | |
| let k : ℕ := dyadicExponent (X+1:ℕ) | |
| let r : ℕ := dyadicExponent (L^B) | |
| let m : ℕ := dyadicExponent y | |
| have hx16 : (16:ℝ) ≤ x := by dsimp [x]; exact_mod_cast hX | |
| have hx1 : 1 ≤ x := by linarith | |
| have hx : 0 < x := by linarith | |
| have hL1 : 1 ≤ L := by dsimp [L,x]; linarith | |
| have hL0 : 0 < L := by linarith | |
| have hB : 0 < B := by dsimp [B]; linarith | |
| have hLB : 1 ≤ L^B := Real.one_le_rpow hL1 hB.le | |
| have hneg : L^(-B) ≤ 1 := by | |
| simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hL1 (neg_nonpos.mpr hB.le) | |
| have hLL : L^B*L^B ≤ x^(1/2:ℝ) := by | |
| calc | |
| _ ≤ x^(1/32:ℝ)*x^(1/32:ℝ) := mul_le_mul hpower hpower (by positivity) (by positivity) | |
| _ = x^(1/16:ℝ) := by rw [←Real.rpow_add hx]; norm_num | |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hx1 (by norm_num) | |
| have heB : -B = -A-10 := by dsimp [B]; ring | |
| have hyeq : y = 2*(X:ℝ)^(1/2:ℝ)*(Real.log (X:ℝ))^(-A-10) := by | |
| simp only [y,heB,L,x] | |
| have hLy : L^B ≤ y := by | |
| have he : L^B*L^(-B)=1 := by rw [←Real.rpow_add hL0]; simp | |
| calc | |
| _ = (L^B*L^B)*L^(-B) := by rw [mul_assoc,he,mul_one] | |
| _ ≤ x^(1/2:ℝ)*L^(-B) := mul_le_mul_of_nonneg_right hLL (by positivity) | |
| _ ≤ y := by dsimp [y]; nlinarith [mul_nonneg (Real.rpow_nonneg hx.le (1/2:ℝ)) (Real.rpow_nonneg hL0.le (-B))] | |
| have hy1 : 1 ≤ y := hLB.trans hLy | |
| have hry : r ≤ m := dyadicExponent_mono hLy | |
| have hrbounds := dyadicUpper_bounds hLB | |
| have hmbounds := dyadicUpper_bounds hy1 | |
| have hsqrt : 4*x^(1/2:ℝ) ≤ x := by | |
| rw [←Real.sqrt_eq_rpow] | |
| have hs : Real.sqrt x ≤ x/4 := by | |
| apply (Real.sqrt_le_left (by positivity)).mpr | |
| nlinarith | |
| linarith | |
| have hymX : 2*y ≤ x := by | |
| calc | |
| _ = 4*x^(1/2:ℝ)*L^(-B) := by dsimp [y]; ring | |
| _ ≤ 4*x^(1/2:ℝ) := (mul_le_mul_of_nonneg_left hneg (by positivity)).trans_eq (mul_one _) | |
| _ ≤ _ := hsqrt | |
| have hmX : 2^m ≤ X := by | |
| have hh : ((2^m:ℕ):ℝ) ≤ (X:ℝ) := hmbounds.2.trans hymX | |
| exact_mod_cast hh | |
| have hymk : y ≤ (X+1:ℕ) := by | |
| have hy0 : 0 ≤ y := by positivity | |
| have hyx : y ≤ x := by linarith | |
| push_cast | |
| change y ≤ x+1 | |
| linarith | |
| have hmk : m ≤ k := dyadicExponent_mono hymk | |
| have hkbounds := vaughanBinaryExponent_bounds (show 2 ≤ X by omega) (show 1 ≤ Real.log (X:ℝ) by linarith) | |
| have hrnm : r+(m-r)=m := Nat.add_sub_of_le hry | |
| refine ⟨T,k,r,m-r,hkbounds.1,?_,?_,?_,?_,?_,?_,hkbounds.2,?_,?_⟩ | |
| · rw [hrnm,←hyeq] | |
| exact hmbounds.1.le | |
| · simpa only [hrnm] using hmX | |
| · rw [hrnm] | |
| calc | |
| _ ≤ 2*y := hmbounds.2 | |
| _ = _ := by rw [hyeq]; ring | |
| · exact hrbounds.1.le | |
| · exact hrbounds.2 | |
| · exact (by exact_mod_cast (Nat.sub_le m r).trans hmk : ((m-r:ℕ):ℝ) ≤ k).trans hkbounds.2 | |
| · exact Nat.floor_le (Real.rpow_nonneg hx.le _) | |
| · have ht := (Nat.lt_floor_add_one (x^(1/16:ℝ))).le | |
| have hh := Real.sqrt_le_sqrt ht | |
| change Real.sqrt (x^(1/16:ℝ)) ≤ Real.sqrt ((T:ℝ)+1) at hh | |
| rw [Real.sqrt_eq_rpow,←Real.rpow_mul hx.le] at hh | |
| norm_num at hh | |
| simpa only [Nat.cast_add,Nat.cast_one] using hh | |
| theorem endpoint_errors_nat_log_bound {A M : ℝ} (hA : 0 < A) (hM : 0 ≤ M) | |
| {X Q : ℕ} (hX : 16 ≤ X) (hL : 2 ≤ Real.log (X:ℝ)) | |
| (hpower : (Real.log (X:ℝ))^(A+10) ≤ (X:ℝ)^(1/32:ℝ)) | |
| (hQ : (Q:ℝ) ≤ 2*(X:ℝ)^(1/2:ℝ)*(Real.log (X:ℝ))^(-A-10)) | |
| (hsmall : ∀ d : ℕ, 0 < d → (d:ℝ) ≤ (Real.log (X:ℝ))^(A+11) → | |
| ∀ χ : DirichletCharacter ℂ d, χ.IsPrimitive → | |
| ‖primeCharacterError (X:ℝ) χ‖ ≤ M*(X:ℝ)*(Real.log (X:ℝ))^(-2*A-14)) : | |
| (∑ q∈Finset.Icc 1 Q,(endpointError (X:ℝ) q:ℝ)) ≤ | |
| (8*M+192024)*(X:ℝ)*(Real.log (X:ℝ))^(-A) := by | |
| obtain ⟨T,k,r,n,hk,hcover,hPX,hP,hRlo,hRhi,hn,hkL,hT,hTroot⟩ := | |
| exists_vaughan_log_parameters hA hX hL hpower | |
| let P : ℕ := 2^(r+n) | |
| let R : ℕ := 2^r | |
| let x : ℝ := X | |
| let L : ℝ := Real.log x | |
| have hx1 : 1 ≤ x := by dsimp [x]; exact_mod_cast (by omega : 1 ≤ X) | |
| have hx : 0 < x := lt_of_lt_of_le zero_lt_one hx1 | |
| have hL1 : 1 ≤ L := by dsimp [L,x]; linarith | |
| have hL0 : 0 < L := lt_of_lt_of_le zero_lt_one hL1 | |
| have hQP : Q ≤ P := by | |
| change Q ≤ 2^(r+n) | |
| exact_mod_cast hQ.trans hcover | |
| have hH : (harmonic P:ℝ) ≤ 2*L := harmonic_le_twice_log hPX hL1 | |
| have hH2 : (harmonic P:ℝ)^2 ≤ 4*L^2 := by nlinarith [harmonic_real_nonneg P] | |
| have hsmallR : (∑ d∈Finset.Icc 1 R,primitiveErrorMass x d/(d.totient:ℝ)) ≤ | |
| (R:ℝ)*(M*x*L^(-2*A-14)) := by | |
| apply small_conductor_mass_le (by positivity) | |
| intro d hd hdR χ hχ | |
| apply hsmall d hd ?_ χ hχ | |
| calc | |
| (d:ℝ) ≤ R := by exact_mod_cast hdR | |
| _ ≤ 2*L^(A+10) := hRhi | |
| _ ≤ L*L^(A+10) := mul_le_mul_of_nonneg_right hL (by positivity) | |
| _ = L^(A+11) := by rw [show A+11=1+(A+10) by ring,Real.rpow_add hL0 1 (A+10),Real.rpow_one] | |
| have hsmallTotal : (∑ d∈Finset.Icc 1 R,primitiveErrorMass x d/(d.totient:ℝ)) * | |
| (harmonic P:ℝ)^2 ≤ 8*M*x*L^(-A) := by | |
| calc | |
| _ ≤ (2*L^(A+10)*(M*x*L^(-2*A-14)))*(4*L^2) := by | |
| apply mul_le_mul (hsmallR.trans (mul_le_mul_of_nonneg_right hRhi (by positivity))) hH2 | |
| (sq_nonneg _) (by positivity) | |
| _ = 8*M*x*L^(-A-2) := by | |
| conv_rhs => rw [show -A-2=(A+10)+(-2*A-14)+2 by ring, | |
| Real.rpow_add hL0,Real.rpow_add hL0,Real.rpow_two] | |
| ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le hL1 (by linarith)) (by positivity) | |
| have hhi := highConductorBlockBound_log_le hA (show 2 ≤ X by omega) hL1 hPX | |
| hP hRlo hkL hT hTroot hpower | |
| have hhighTotal : ((n:ℝ)*highConductorBlockBound P X T R k)*(harmonic P:ℝ)^2 ≤ | |
| 192000*x*L^(-A) := by | |
| calc | |
| _ ≤ ((6*L)*(8000*x*L^(-A-3)))*(4*L^2) := by | |
| apply mul_le_mul (mul_le_mul hn hhi (highConductorBlockBound_nonneg _ _ _ _ _) (by positivity)) | |
| hH2 (sq_nonneg _) (by positivity) | |
| _ = _ := by | |
| conv_rhs => rw [show -A=(-A-3)+1+2 by ring,Real.rpow_add hL0,Real.rpow_add hL0, | |
| Real.rpow_one,Real.rpow_two] | |
| ring | |
| have hlog2 : (Nat.log 2 X:ℝ) ≤ 6*L := by | |
| have hmono : Nat.log 2 X ≤ dyadicExponent (X+1:ℕ) := by | |
| unfold dyadicExponent | |
| rw [Nat.floor_natCast] | |
| exact (Nat.log_mono_right (Nat.le_succ X)).trans (Nat.le_succ _) | |
| exact (by exact_mod_cast hmono : (Nat.log 2 X:ℝ) ≤ (dyadicExponent (X+1:ℕ):ℝ)).trans | |
| (vaughanBinaryExponent_bounds (show 2 ≤ X by omega) hL1).2 | |
| have hlogP : Real.log (P:ℝ) ≤ L := log_nat_monotone hPX | |
| have hcost : (P:ℝ)*(Nat.log 2 X:ℝ)*Real.log P ≤ 24*x*L^(-A) := by | |
| calc | |
| _ ≤ (4*x^(1/2:ℝ)*L^(-A-10))*(6*L)*L := by | |
| gcongr | |
| _ = 24*x^(1/2:ℝ)*L^(-A-8) := by | |
| rw [show -A-8=(-A-10)+1+1 by ring,Real.rpow_add hL0,Real.rpow_add hL0,Real.rpow_one] | |
| ring | |
| _ ≤ _ := by | |
| have hsx : x^(1/2:ℝ) ≤ x := by | |
| simpa only [Real.rpow_one] using (Real.rpow_le_rpow_of_exponent_le hx1 | |
| (by norm_num : (1/2:ℝ) ≤ 1)) | |
| apply mul_le_mul (mul_le_mul_of_nonneg_left hsx (by norm_num : (0:ℝ) ≤ 24)) | |
| (Real.rpow_le_rpow_of_exponent_le hL1 (by linarith)) (by positivity) (by positivity) | |
| have hfinite := endpoint_errors_finite_vaughan_reduction_of_le X T k r n Q hk hQP | |
| apply hfinite.trans | |
| change ((∑ d∈Finset.Icc 1 R,primitiveErrorMass x d/(d.totient:ℝ))+ | |
| (n:ℝ)*highConductorBlockBound P X T R k)*(harmonic P:ℝ)^2+ | |
| (P:ℝ)*(Nat.log 2 X:ℝ)*Real.log P ≤ (8*M+192024)*x*L^(-A) | |
| rw [add_mul] | |
| nlinarith [hsmallTotal,hhighTotal,hcost] | |
| theorem endpoint_errors_real_log_bound {A M x : ℝ} (hA : 0 < A) (hM : 0 ≤ M) | |
| (hx : 0 ≤ x) (hX : 16 ≤ ⌊x⌋₊) (hL : 2 ≤ Real.log (⌊x⌋₊:ℝ)) | |
| (hpower : (Real.log (⌊x⌋₊:ℝ))^(A+10) ≤ (⌊x⌋₊:ℝ)^(1/32:ℝ)) | |
| (hsmall : ∀ d : ℕ, 0 < d → (d:ℝ) ≤ (Real.log (⌊x⌋₊:ℝ))^(A+11) → | |
| ∀ χ : DirichletCharacter ℂ d, χ.IsPrimitive → | |
| ‖primeCharacterError (⌊x⌋₊:ℝ) χ‖ ≤ M*(⌊x⌋₊:ℝ)*(Real.log (⌊x⌋₊:ℝ))^(-2*A-14)) : | |
| (∑ q∈Finset.Icc 1 ⌊x^(1/2:ℝ)*(Real.log x)^(-(A+10))⌋₊,(endpointError x q:ℝ)) ≤ | |
| ((8*M+192028)*(2:ℝ)^A)*x*(Real.log x)^(-A) := by | |
| let X : ℕ := ⌊x⌋₊ | |
| let Q : ℕ := ⌊x^(1/2:ℝ)*(Real.log x)^(-(A+10))⌋₊ | |
| let L : ℝ := Real.log (X:ℝ) | |
| have hX16 : (16:ℝ) ≤ X := by | |
| change (16:ℝ) ≤ (⌊x⌋₊:ℝ) | |
| exact_mod_cast hX | |
| have hX1 : (1:ℝ) ≤ X := by linarith | |
| have hX0 : (0:ℝ) < X := by linarith | |
| have hfloor : (X:ℝ) ≤ x := Nat.floor_le hx | |
| have hceil : x < (X:ℝ)+1 := Nat.lt_floor_add_one x | |
| have hL1 : 1 ≤ L := by dsimp [L,X]; linarith | |
| have hL0 : 0 < L := by linarith | |
| have hloglo : L ≤ Real.log x := Real.log_le_log hX0 hfloor | |
| have hlogx0 : 0 < Real.log x := hL0.trans_le hloglo | |
| have hloghi : Real.log x ≤ 2*L := | |
| calc | |
| _ ≤ Real.log (X+1:ℕ) := Real.log_le_log (hX0.trans_le hfloor) (by push_cast; linarith) | |
| _ ≤ _ := log_add_one_le_twice (by omega) | |
| have hroot : x^(1/2:ℝ) ≤ 2*(X:ℝ)^(1/2:ℝ) := by | |
| rw [←Real.sqrt_eq_rpow,←Real.sqrt_eq_rpow] | |
| calc | |
| _ ≤ Real.sqrt (4*(X:ℝ)) := Real.sqrt_le_sqrt (by linarith) | |
| _ = _ := by rw [Real.sqrt_mul (by norm_num : (0:ℝ) ≤ 4)]; norm_num | |
| have hQ : (Q:ℝ) ≤ 2*(X:ℝ)^(1/2:ℝ)*L^(-A-10) := by | |
| calc | |
| _ ≤ x^(1/2:ℝ)*(Real.log x)^(-(A+10)) := Nat.floor_le (by positivity) | |
| _ ≤ (2*(X:ℝ)^(1/2:ℝ))*L^(-(A+10)) := mul_le_mul hroot | |
| (Real.rpow_le_rpow_of_nonpos hL0 hloglo (by linarith)) (by positivity) (by positivity) | |
| _ = _ := by congr 1; congr 1; ring | |
| have hneg : L^(-A-10) ≤ 1 := by | |
| simpa only [Real.rpow_zero] using Real.rpow_le_rpow_of_exponent_le hL1 | |
| (show -A-10 ≤ 0 by linarith) | |
| have hQX : Q ≤ X := by | |
| have hrootsmall : 2*(X:ℝ)^(1/2:ℝ) ≤ X := by | |
| rw [←Real.sqrt_eq_rpow] | |
| have hs : Real.sqrt (X:ℝ) ≤ (X:ℝ)/2 := by | |
| apply (Real.sqrt_le_left (by positivity)).mpr | |
| nlinarith | |
| linarith | |
| have hh₁ : 2*(X:ℝ)^(1/2:ℝ)*L^(-A-10) ≤ 2*(X:ℝ)^(1/2:ℝ) := by | |
| simpa only [mul_one] using mul_le_mul_of_nonneg_left hneg | |
| (by positivity : 0 ≤ 2*(X:ℝ)^(1/2:ℝ)) | |
| have hh : (Q:ℝ) ≤ X := hQ.trans (hh₁.trans hrootsmall) | |
| exact_mod_cast hh | |
| have hnat := endpoint_errors_nat_log_bound hA hM hX hL hpower hQ hsmall | |
| have hH : (harmonic Q:ℝ) ≤ 2*L := harmonic_le_twice_log hQX hL1 | |
| have habs : L^2 ≤ (X:ℝ)*L^(-A) := by | |
| convert rpow_absorption_of_exponents (c:=2) hX1 hL1 hpower | |
| (show A+2 ≤ A+10 by linarith) (show (1/32:ℝ) ≤ 1 by norm_num) using 1 <;> norm_num | |
| have htotal : (∑ q∈Finset.Icc 1 Q,(endpointError x q:ℝ)) ≤ | |
| (8*M+192028)*(X:ℝ)*L^(-A) := by | |
| have hh := sum_endpointError_le_floor_add_harmonic hx Q | |
| have hH2 : (harmonic Q:ℝ)^2 ≤ 4*L^2 := by nlinarith [harmonic_real_nonneg Q] | |
| nlinarith [hnat,hh,habs,hH2] | |
| have hlogscale : L^(-A) ≤ (2:ℝ)^A*(Real.log x)^(-A) := by | |
| calc | |
| _ ≤ (Real.log x/2)^(-A) := Real.rpow_le_rpow_of_nonpos (by linarith) | |
| (by linarith) (by linarith) | |
| _ = _ := by | |
| rw [Real.div_rpow (by linarith : 0 ≤ Real.log x) (by norm_num : (0:ℝ) ≤ 2), | |
| Real.rpow_neg (by norm_num : (0:ℝ) ≤ 2),div_inv_eq_mul] | |
| ring | |
| apply htotal.trans | |
| calc | |
| _ ≤ (8*M+192028)*x*((2:ℝ)^A*(Real.log x)^(-A)) := by gcongr | |
| _ = _ := by ring | |
| theorem bombieriVinogradov_of_siegelWalfisz (hs : SiegelWalfisz) : BombieriVinogradov := by | |
| intro A hA | |
| obtain ⟨M,hM,x₀,hx₀,hSW⟩ := hs (2*A+14) (A+11) (by linarith) (by linarith) | |
| refine ⟨A+10,by linarith,(8*M+192028)*(2:ℝ)^A,by positivity,?_⟩ | |
| have hf : Tendsto (fun x : ℝ => (⌊x⌋₊:ℝ)) atTop atTop := | |
| tendsto_natCast_atTop_atTop.comp tendsto_nat_floor_atTop | |
| have he : ∀ᶠ x : ℝ in atTop, | |
| (∑ q∈Finset.Icc 1 ⌊x^(1/2:ℝ)*(Real.log x)^(-(A+10))⌋₊,(endpointError x q:ℝ)) ≤ | |
| ((8*M+192028)*(2:ℝ)^A)*x*(Real.log x)^(-A) := by | |
| filter_upwards [eventually_ge_atTop (0:ℝ), | |
| tendsto_nat_floor_atTop.eventually (eventually_ge_atTop (16:ℕ)), | |
| (Real.tendsto_log_atTop.comp hf).eventually_ge_atTop 2, | |
| hf.eventually_ge_atTop x₀, | |
| hf.eventually (eventually_log_rpow_le_rpow (A+10) (by norm_num : (0:ℝ)<1/32))] | |
| with x hx hX hL hxsw hp | |
| apply endpoint_errors_real_log_bound hA hM.le hx hX hL hp | |
| intro d hd hdL χ hχ | |
| convert hSW (⌊x⌋₊:ℝ) hxsw d hd hdL χ hχ using 1 | |
| congr 1 | |
| congr 1 | |
| ring | |
| obtain ⟨z,hz⟩ := eventually_atTop.mp he | |
| refine ⟨max z 2,by linarith [le_max_right z (2:ℝ)],?_⟩ | |
| intro x hx | |
| exact hz x ((le_max_left z (2:ℝ)).trans hx) | |
| open Complex Filter Topology | |
| noncomputable def poleRemovedL {q : ℕ} [NeZero q] (χ : DirichletCharacter ℂ q) : ℂ → ℂ := | |
| if χ = 1 then DirichletCharacter.LFunctionTrivChar₁ q else χ.LFunction | |
| noncomputable def primeLogDerivativeRegular {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) (s : ℂ) : ℂ := | |
| -deriv (poleRemovedL χ) s / poleRemovedL χ s | |
| theorem poleRemovedL_differentiable {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) : Differentiable ℂ (poleRemovedL χ) := by | |
| unfold poleRemovedL | |
| split_ifs with hχ | |
| · exact DirichletCharacter.differentiable_LFunctionTrivChar₁ q | |
| · exact DirichletCharacter.differentiable_LFunction hχ | |
| theorem poleRemovedL_ne_zero {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) {s : ℂ} (hs : 1 ≤ s.re) : | |
| poleRemovedL χ s ≠ 0 := by | |
| unfold poleRemovedL | |
| split_ifs with hχ | |
| · rcases eq_or_ne s 1 with rfl | hs1 | |
| · exact DirichletCharacter.LFunctionTrivChar₁_apply_one_ne_zero q | |
| · rw [DirichletCharacter.LFunctionTrivChar₁, Function.update_of_ne hs1] | |
| refine mul_ne_zero (sub_ne_zero.mpr hs1) ?_ | |
| exact DirichletCharacter.LFunction_ne_zero_of_one_le_re 1 (.inr hs1) hs | |
| · exact DirichletCharacter.LFunction_ne_zero_of_one_le_re χ (.inl hχ) hs | |
| theorem primeLogDerivativeRegular_analyticAt {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) {s : ℂ} (hs : poleRemovedL χ s ≠ 0) : | |
| AnalyticAt ℂ (primeLogDerivativeRegular χ) s := by | |
| have h := (poleRemovedL_differentiable χ).analyticAt s | |
| exact h.deriv.neg.div h hs | |
| theorem primeLogDerivativeRegular_eq {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) {s : ℂ} (hs : 1 < s.re) : | |
| primeLogDerivativeRegular χ s = | |
| LSeries (fun n : ℕ => χ (n : ZMod q) * (ArithmeticFunction.vonMangoldt n : ℂ)) s - | |
| (if χ = 1 then 1 / (s - 1) else 0) := by | |
| have hs1 : s ≠ 1 := by intro h; simp [h] at hs | |
| have hL := DirichletCharacter.LSeries_twist_vonMangoldt_eq χ hs | |
| change LSeries (fun n : ℕ => χ (n : ZMod q) * (ArithmeticFunction.vonMangoldt n : ℂ)) s = | |
| -deriv (LSeries (fun n : ℕ => χ (n : ZMod q))) s / | |
| LSeries (fun n : ℕ => χ (n : ZMod q)) s at hL | |
| rw [hL, ← DirichletCharacter.LFunction_eq_LSeries χ hs, | |
| ← DirichletCharacter.deriv_LFunction_eq_deriv_LSeries χ hs] | |
| by_cases hχ : χ = 1 | |
| · subst χ | |
| have hn := DirichletCharacter.LFunction_ne_zero_of_one_le_re | |
| (1 : DirichletCharacter ℂ q) (.inr hs1) hs.le | |
| simp only [primeLogDerivativeRegular, poleRemovedL, ↓reduceIte] | |
| rw [DirichletCharacter.deriv_LFunctionTrivChar₁_apply_of_ne_one q hs1, | |
| DirichletCharacter.LFunctionTrivChar₁, Function.update_of_ne hs1] | |
| change -((s-1)*deriv (DirichletCharacter.LFunction (1 : DirichletCharacter ℂ q)) s + | |
| DirichletCharacter.LFunction (1 : DirichletCharacter ℂ q) s) / | |
| ((s-1)*DirichletCharacter.LFunction (1 : DirichletCharacter ℂ q) s) = _ | |
| field_simp | |
| ring | |
| · simp [primeLogDerivativeRegular, poleRemovedL, hχ] | |
| theorem poleRemovedL_fixed_height_strip {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) (T : ℝ) : | |
| ∃ δ : ℝ, 0 < δ ∧ ∀ s : ℂ, 1-δ ≤ s.re → |s.im| ≤ T → | |
| poleRemovedL χ s ≠ 0 := by | |
| let K : Set ℂ := (fun t : ℝ => (1 : ℂ)+(t:ℂ)*I) '' Set.Icc (-T) T | |
| have hK : IsCompact K := isCompact_Icc.image (by fun_prop) | |
| have hO : IsOpen {s : ℂ | poleRemovedL χ s ≠ 0} := | |
| isOpen_ne_fun (poleRemovedL_differentiable χ).continuous continuous_const | |
| have hKO : K ⊆ {s : ℂ | poleRemovedL χ s ≠ 0} := by | |
| rintro s ⟨t, ht, rfl⟩ | |
| exact poleRemovedL_ne_zero χ (by simp) | |
| obtain ⟨ε, hε, hεK⟩ := hK.exists_thickening_subset_open hO hKO | |
| refine ⟨ε/2, by positivity, ?_⟩ | |
| intro s hs ht | |
| by_cases hs1 : 1 ≤ s.re | |
| · exact poleRemovedL_ne_zero χ hs1 | |
| apply hεK | |
| apply Metric.mem_thickening_iff.mpr | |
| refine ⟨1+(s.im:ℂ)*I, ⟨s.im, abs_le.mp ht, rfl⟩, ?_⟩ | |
| have hd : dist s (1+(s.im:ℂ)*I) = |s.re-1| := by | |
| rw [dist_eq_norm] | |
| have he : s-(1+(s.im:ℂ)*I) = ((s.re-1:ℝ):ℂ) := by | |
| apply Complex.ext <;> simp | |
| rw [he, Complex.norm_real, Real.norm_eq_abs] | |
| rw [hd, abs_of_nonpos (by linarith)] | |
| linarith | |
| theorem primeLogDerivativeRegular_fixed_height_bound {q : ℕ} [NeZero q] | |
| (χ : DirichletCharacter ℂ q) (T : ℝ) : | |
| ∃ δ M : ℝ, 0 < δ ∧ 0 < M ∧ ∀ s : ℂ, | |
| 1-δ ≤ s.re → s.re ≤ 2 → |s.im| ≤ T → | |
| AnalyticAt ℂ (primeLogDerivativeRegular χ) s ∧ | |
| ‖primeLogDerivativeRegular χ s‖ ≤ M := by | |
| obtain ⟨δ, hδ, hn⟩ := poleRemovedL_fixed_height_strip χ T | |
| let K : Set ℂ := (fun p : ℝ × ℝ => (p.1:ℂ)+(p.2:ℂ)*I) '' | |
| (Set.Icc (1-δ) 2 ×ˢ Set.Icc (-T) T) | |
| have hK : IsCompact K := (isCompact_Icc.prod isCompact_Icc).image (by fun_prop) | |
| have hmem (s : ℂ) : s ∈ K ↔ 1-δ ≤ s.re ∧ s.re ≤ 2 ∧ |s.im| ≤ T := by | |
| constructor | |
| · rintro ⟨⟨u,v⟩, ⟨hu, hv⟩, rfl⟩ | |
| simpa [abs_le, and_assoc] using And.intro hu hv | |
| · rintro ⟨hs1, hs2, hs3⟩ | |
| refine ⟨(s.re,s.im), ⟨⟨hs1,hs2⟩,abs_le.mp hs3⟩, ?_⟩ | |
| exact Complex.re_add_im s | |
| have hc : ContinuousOn (primeLogDerivativeRegular χ) K := by | |
| intro s hs | |
| obtain ⟨hs1, hs2, hs3⟩ := (hmem s).mp hs | |
| exact (primeLogDerivativeRegular_analyticAt χ (hn s hs1 hs3)).continuousAt.continuousWithinAt | |
| obtain ⟨M, hM⟩ := hK.exists_bound_of_continuousOn hc | |
| refine ⟨δ, max M 1, hδ, lt_of_lt_of_le zero_lt_one (le_max_right _ _), ?_⟩ | |
| intro s hs1 hs2 hs3 | |
| exact ⟨primeLogDerivativeRegular_analyticAt χ (hn s hs1 hs3), | |
| (hM s ((hmem s).mpr ⟨hs1,hs2,hs3⟩)).trans (le_max_left _ _)⟩ | |
| end LargePrimeGaps | |
| end OAI | |