| {"name":"ZetaSum_aux1_5d","declaration":"theorem ZetaSum_aux1_5d {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : IntervalIntegrable (fun u => |↑⌊u⌋ + 1 / 2 - u| / u ^ (s.re + 1)) MeasureTheory.volume a b"} | |
| {"name":"ZetaSum_aux1","declaration":"theorem ZetaSum_aux1 {a : ℕ} {b : ℕ} {s : ℂ} (s_ne_one : s ≠ 1) (s_ne_zero : s ≠ 0) (ha : a ∈ Set.Ioo 0 b) : (Finset.sum (Finset.Ioc ↑a ↑b) fun n => 1 / ↑n ^ s) =\n (↑b ^ (1 - s) - ↑a ^ (1 - s)) / (1 - s) + 1 / 2 * (1 / ↑b ^ s) - 1 / 2 * (1 / ↑a ^ s) +\n s * ∫ (x : ℝ) in ↑a..↑b, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1))"} | |
| {"name":"xpos_of_uIcc","declaration":"theorem xpos_of_uIcc {a : ℕ} {b : ℕ} (ha : a ∈ Set.Ioo 0 b) {x : ℝ} (x_in : x ∈ Set.uIcc ↑a ↑b) : 0 < x"} | |
| {"name":"ZetaSum_aux1_5","declaration":"theorem ZetaSum_aux1_5 {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : ∫ (x : ℝ) in a..b, |↑⌊x⌋ + 1 / 2 - x| / x ^ (s.re + 1) ≤ ∫ (x : ℝ) in a..b, 1 / x ^ (s.re + 1)"} | |
| {"name":"riemannzeta","declaration":"/-- We use `ζ` to denote the Rieman zeta function and `ζ₀` to denote the alternative\nRieman zeta function.. -/\ndef riemannzeta : Lean.ParserDescr"} | |
| {"name":"Complex.cpow_inv_tendsto","declaration":"theorem Complex.cpow_inv_tendsto {s : ℂ} (hs : 0 < s.re) : Filter.Tendsto (fun x => (↑x ^ s)⁻¹) Filter.atTop (nhds 0)"} | |
| {"name":"ZetaBnd_aux2","declaration":"theorem ZetaBnd_aux2 {n : ℕ} {t : ℝ} {A : ℝ} {σ : ℝ} (Apos : 0 < A) (σpos : 0 < σ) (n_le_t : ↑n ≤ |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : ‖↑n ^ (-(↑σ + ↑t * Complex.I))‖ ≤ (↑n)⁻¹ * Real.exp A"} | |
| {"name":"Ioi_union_Iio_mem_cocompact","declaration":"theorem Ioi_union_Iio_mem_cocompact {a : ℝ} (ha : 0 ≤ a) : Set.Ioi a ∪ Set.Iio (-a) ∈ Filter.cocompact ℝ"} | |
| {"name":"Zeta0EqZeta","declaration":"theorem Zeta0EqZeta {N : ℕ} (N_pos : 0 < N) {s : ℂ} (reS_pos : 0 < s.re) (s_ne_one : s ≠ 1) : riemannZeta0 N s = riemannZeta s"} | |
| {"name":"one_div_cpow_eq_cpow_neg","declaration":"theorem one_div_cpow_eq_cpow_neg (x : ℂ) (s : ℂ) : 1 / x ^ s = x ^ (-s)"} | |
| {"name":"ZetaInvBound1","declaration":"theorem ZetaInvBound1 {σ : ℝ} {t : ℝ} (σ_gt : 1 < σ) : 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ ‖riemannZeta ↑σ‖ ^ (3 / 4) * ‖riemannZeta (↑σ + 2 * ↑t * Complex.I)‖ ^ (1 / 4)"} | |
| {"name":"integrability_aux₀","declaration":"theorem integrability_aux₀ {a : ℝ} {b : ℝ} : ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict (Set.uIcc a b), ‖↑⌊x⌋‖ ≤ max ‖a‖ ‖b‖ + 1"} | |
| {"name":"finsetSum_tendsto_tsum","declaration":"theorem finsetSum_tendsto_tsum {N : ℕ} {f : ℕ → ℂ} (hf : Summable f) : Filter.Tendsto (fun k => Finset.sum (Finset.Ioc N k) fun n => f n) Filter.atTop (nhds (∑' (n : ℕ), f (n + N)))"} | |
| {"name":"Nat.self_div_floor_bound","declaration":"theorem Nat.self_div_floor_bound {t : ℝ} (t_ge : 1 ≤ |t|) : |t| / ↑⌊|t|⌋₊ ∈ Set.Icc 1 2"} | |
| {"name":"Summable_rpow","declaration":"theorem Summable_rpow {s : ℂ} (s_re_gt : 1 < s.re) : Summable fun n => 1 / ↑n ^ s"} | |
| {"name":"ZetaSum_aux1_1'","declaration":"theorem ZetaSum_aux1_1' {a : ℝ} {b : ℝ} {x : ℝ} (apos : 0 < a) (hx : x ∈ Set.Icc a b) : 0 < x"} | |
| {"name":"Tendsto_nhdsWithin_punctured_map_add","declaration":"theorem Tendsto_nhdsWithin_punctured_map_add {f : ℝ → ℝ} (a : ℝ) (x : ℝ) (f_mono : StrictMono f) (f_iso : Isometry f) : Filter.Tendsto (fun y => f y + a) (nhdsWithin x (Set.Ioi x)) (nhdsWithin (f x + a) (Set.Ioi (f x + a)))"} | |
| {"name":"add_le_add_le_add_le_add","declaration":"theorem add_le_add_le_add_le_add {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} {c : α} {d : α} {e : α} {f : α} {g : α} {h : α} (h₁ : a ≤ b) (h₂ : c ≤ d) (h₃ : e ≤ f) (h₄ : g ≤ h) : a + c + e + g ≤ b + d + f + h"} | |
| {"name":"ZetaInvBound2","declaration":"theorem ZetaInvBound2 {σ : ℝ} (hσ : σ ∈ Set.Ioc 1 2) : (fun t => 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖) =O[Filter.cocompact ℝ] fun t =>\n (σ - 1) ^ (-3 / 4) * Real.log |t| ^ (1 / 4)"} | |
| {"name":"Tendsto_nhdsWithin_punctured_add","declaration":"theorem Tendsto_nhdsWithin_punctured_add (a : ℝ) (x : ℝ) : Filter.Tendsto (fun y => y + a) (nhdsWithin x (Set.Ioi x)) (nhdsWithin (x + a) (Set.Ioi (x + a)))"} | |
| {"name":"sum_eq_int_deriv_aux","declaration":"theorem sum_eq_int_deriv_aux {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_le_kpOne : b ≤ ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc ⌊a⌋ ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑⌊a⌋ + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) * deriv φ x"} | |
| {"name":"add_le_add_le_add","declaration":"theorem add_le_add_le_add {α : Type u_1} [Add α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α} {b : α} {c : α} {d : α} {e : α} {f : α} (h₁ : a ≤ b) (h₂ : c ≤ d) (h₃ : e ≤ f) : a + c + e ≤ b + d + f"} | |
| {"name":"ContDiffOn.hasDeriv_deriv","declaration":"theorem ContDiffOn.hasDeriv_deriv {φ : ℝ → ℂ} {s : Set ℝ} (φDiff : ContDiffOn ℝ 1 φ s) {x : ℝ} (x_in_s : s ∈ nhds x) : HasDerivAt φ (deriv φ x) x"} | |
| {"name":"UpperBnd_aux2","declaration":"theorem UpperBnd_aux2 {A : ℝ} {σ : ℝ} {t : ℝ} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : |t| ^ (1 - σ) ≤ Real.exp A"} | |
| {"name":"ZetaDerivUpperBnd","declaration":"theorem ZetaDerivUpperBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Icc (1 - A / Real.log |t|) 2 → ‖deriv riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 2"} | |
| {"name":"Finset_coe_Nat_Int","declaration":"theorem Finset_coe_Nat_Int (f : ℤ → ℂ) (m : ℕ) (n : ℕ) : (Finset.sum (Finset.Ioc m n) fun x => f ↑x) = Finset.sum (Finset.Ioc ↑m ↑n) fun x => f x"} | |
| {"name":"div_rpow_eq_rpow_div_neg","declaration":"theorem div_rpow_eq_rpow_div_neg {x : ℝ} {y : ℝ} {s : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x ^ s / y ^ s = (y / x) ^ (-s)"} | |
| {"name":"ZetaSum_aux1φderiv","declaration":"theorem ZetaSum_aux1φderiv {s : ℂ} (s_ne_zero : s ≠ 0) {x : ℝ} (xpos : 0 < x) : deriv (fun t => 1 / ↑t ^ s) x = (fun x => -s * ↑x ^ (-(s + 1))) x"} | |
| {"name":"interval_induction","declaration":"theorem interval_induction (P : ℝ → ℝ → Prop) (base : ∀ (a b : ℝ) (k : ℤ), ↑k ≤ a → a < b → b ≤ ↑k + 1 → P a b) (step : ∀ (a : ℝ) (k : ℤ) (b : ℝ), a < ↑k → ↑k < b → P a ↑k → P (↑k) b → P a b) (a : ℝ) (b : ℝ) (hab : a < b) : P a b"} | |
| {"name":"riemannZeta0_zero_aux","declaration":"theorem riemannZeta0_zero_aux (N : ℕ) (Npos : 0 < N) : (Finset.sum (Finset.Ico 0 N) fun x => (↑x)⁻¹) = Finset.sum (Finset.Ico 1 N) fun x => (↑x)⁻¹"} | |
| {"name":"ZetaSum_aux1a","declaration":"theorem ZetaSum_aux1a {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : ‖∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (s + 1)‖ ≤ (a ^ (-s.re) - b ^ (-s.re)) / s.re"} | |
| {"name":"ct_aux1","declaration":"def ct_aux1 : ℕ"} | |
| {"name":"tendsto_coe_atTop","declaration":"theorem tendsto_coe_atTop : Filter.Tendsto (fun n => ↑n) Filter.atTop Filter.atTop"} | |
| {"name":"ZetaSum_aux1_4","declaration":"theorem ZetaSum_aux1_4 {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} : ∫ (x : ℝ) in a..b, ‖(↑⌊x⌋ + ↑1 / 2 - ↑x) / ↑x ^ (s + 1)‖ = ∫ (x : ℝ) in a..b, |↑⌊x⌋ + 1 / 2 - x| / x ^ (s + 1).re"} | |
| {"name":"ZetaSum_aux1_5b","declaration":"theorem ZetaSum_aux1_5b {a : ℝ} {b : ℝ} (apos : 0 < a) (a_lt_b : a < b) {s : ℂ} (σpos : 0 < s.re) : IntervalIntegrable (fun u => 1 / u ^ (s.re + 1)) MeasureTheory.volume a b"} | |
| {"name":"riemannzeta0","declaration":"def riemannzeta0 : Lean.ParserDescr"} | |
| {"name":"LinearDerivative_ofReal","declaration":"theorem LinearDerivative_ofReal (x : ℝ) (a : ℂ) (b : ℂ) : HasDerivAt (fun t => a * ↑t + b) a x"} | |
| {"name":"interval_induction_aux_int","declaration":"theorem interval_induction_aux_int (n : ℕ) (P : ℝ → ℝ → Prop) : (∀ (a b : ℝ) (k : ℤ), ↑k ≤ a → a < b → b ≤ ↑k + 1 → P a b) →\n (∀ (a : ℝ) (k : ℤ) (c : ℝ), a < ↑k → ↑k < c → P a ↑k → P (↑k) c → P a c) → ∀ (a b : ℝ), a < b → ↑n = ⌊b⌋ - ⌊a⌋ → P a b"} | |
| {"name":"UpperBnd_aux3","declaration":"theorem UpperBnd_aux3 {A : ℝ} {C : ℝ} {σ : ℝ} {t : ℝ} (Apos : 0 < A) (A_lt_one : A < 1) {N : ℕ} (Npos : 0 < N) (σ_ge : 1 - A / Real.log |t| ≤ σ) (t_ge : 3 < |t|) (N_le_t : ↑N ≤ |t|) (hC : 2 ≤ C) : ‖Finset.sum (Finset.range N) fun n => ↑n ^ (-(↑σ + ↑t * Complex.I))‖ ≤ Real.exp A * C * Real.log |t|"} | |
| {"name":"Finset.Ioc_diff_Ioc","declaration":"/-- ** Partial summation ** (TODO : Add to Mathlib). -/\ntheorem Finset.Ioc_diff_Ioc {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a : α} {b : α} {c : α} [DecidableEq α] (hb : b ∈ Finset.Icc a c) : Finset.Ioc a b = Finset.Ioc a c \\ Finset.Ioc b c"} | |
| {"name":"ZetaSum_aux1_5c","declaration":"theorem ZetaSum_aux1_5c {a : ℝ} {b : ℝ} {s : ℂ} : let g := fun u => |↑⌊u⌋ + 1 / 2 - u| / u ^ (s.re + 1);\nMeasureTheory.AEStronglyMeasurable g (MeasureTheory.volume.restrict (Ι a b))"} | |
| {"name":"lt_abs_mem_cocompact","declaration":"theorem lt_abs_mem_cocompact {a : ℝ} (ha : 0 ≤ a) : {t | a < |t|} ∈ Filter.cocompact ℝ"} | |
| {"name":"div_rpow_neg_eq_rpow_div","declaration":"theorem div_rpow_neg_eq_rpow_div {x : ℝ} {y : ℝ} {s : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x ^ (-s) / y ^ (-s) = (y / x) ^ s"} | |
| {"name":"C_aux1","declaration":"def C_aux1 : ℕ"} | |
| {"name":"div_cpow_eq_cpow_neg","declaration":"theorem div_cpow_eq_cpow_neg (a : ℂ) (x : ℂ) (s : ℂ) : a / x ^ s = a * x ^ (-s)"} | |
| {"name":"sum_eq_int_deriv","declaration":"theorem sum_eq_int_deriv {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (a_lt_b : a < b) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc ⌊a⌋ ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑⌊a⌋ + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑⌊x⌋ + 1 / 2 - ↑x) * deriv φ x"} | |
| {"name":"ContDiffOn.continuousOn_deriv","declaration":"theorem ContDiffOn.continuousOn_deriv {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (φDiff : ContDiffOn ℝ 1 φ (Set.uIoo a b)) : ContinuousOn (deriv φ) (Set.uIoo a b)"} | |
| {"name":"Finset.Ioc_eq_Ico","declaration":"theorem Finset.Ioc_eq_Ico (M : ℕ) (N : ℕ) : Finset.Ioc N M = Finset.Ico (N + 1) (M + 1)"} | |
| {"name":"isPathConnected_aux","declaration":"theorem isPathConnected_aux : IsPathConnected {z | z ≠ 1 ∧ 0 < z.re}"} | |
| {"name":"ZetaBnd_aux1","declaration":"theorem ZetaBnd_aux1 (N : ℕ) (Npos : 1 ≤ N) {σ : ℝ} (hσ : σ ∈ Set.Ioc 0 2) (t : ℝ) (ht : ↑ct_aux1 < |t|) : ‖(↑σ + ↑t * Complex.I) * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (↑σ + ↑t * Complex.I + 1)‖ ≤\n ↑C_aux1 * |t| * ↑N ^ (-σ) / σ"} | |
| {"name":"ZetaSum_aux1_3b","declaration":"theorem ZetaSum_aux1_3b (x : ℝ) : ↑⌊x⌋ + 1 / 2 - x ≤ 1 / 2"} | |
| {"name":"UpperBnd_aux5","declaration":"theorem UpperBnd_aux5 {σ : ℝ} {t : ℝ} (t_ge : 3 < |t|) (σ_le : σ ≤ 2) : (|t| / ↑⌊|t|⌋₊) ^ σ ≤ 4"} | |
| {"name":"tsum_eq_partial_add_tail","declaration":"theorem tsum_eq_partial_add_tail {N : ℕ} (f : ℕ → ℂ) (hf : Summable f) : ∑' (n : ℕ), f n = (Finset.sum (Finset.Ico 0 N) fun n => f n) + ∑' (n : ℕ), f (n + N)"} | |
| {"name":"integrability_aux₁","declaration":"theorem integrability_aux₁ {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => ↑⌊x⌋) MeasureTheory.volume a b"} | |
| {"name":"Zeta_diff_Bnd","declaration":"theorem Zeta_diff_Bnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ₁ σ₂ t : ℝ),\n 3 < |t| →\n 1 - A / Real.log |t| ≤ σ₁ →\n σ₂ ≤ 2 →\n σ₁ < σ₂ →\n ‖riemannZeta (↑σ₂ + ↑t * Complex.I) - riemannZeta (↑σ₁ + ↑t * Complex.I)‖ ≤\n C * Real.log |t| ^ 2 * (σ₂ - σ₁)"} | |
| {"name":"integral_deriv_mul_eq_sub'","declaration":"theorem integral_deriv_mul_eq_sub' {A : Type u_1} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] {a : ℝ} {b : ℝ} {u : ℝ → A} {v : ℝ → A} {u' : ℝ → A} {v' : ℝ → A} (hu : ∀ x ∈ Set.uIcc a b, HasDerivWithinAt u (u' x) (Set.uIcc a b) x) (hv : ∀ x ∈ Set.uIcc a b, HasDerivWithinAt v (v' x) (Set.uIcc a b) x) (hu' : IntervalIntegrable u' MeasureTheory.volume a b) (hv' : IntervalIntegrable v' MeasureTheory.volume a b) : ∫ (x : ℝ) in a..b, u' x * v x + u x * v' x = u b * v b - u a * v a"} | |
| {"name":"ZetaSum_aux1_1","declaration":"theorem ZetaSum_aux1_1 {a : ℝ} {b : ℝ} {x : ℝ} (apos : 0 < a) (a_lt_b : a < b) (hx : x ∈ Set.uIcc a b) : 0 < x"} | |
| {"name":"ZetaSum_aux1_3","declaration":"theorem ZetaSum_aux1_3 (x : ℝ) : |↑⌊x⌋ + 1 / 2 - x| ≤ 1 / 2"} | |
| {"name":"HolomorphicOn_riemannZeta0","declaration":"theorem HolomorphicOn_riemannZeta0 {N : ℕ} (N_pos : 0 < N) : HolomorphicOn (riemannZeta0 N) {s | s ≠ 1 ∧ 0 < s.re}"} | |
| {"name":"sum_eq_int_deriv_aux1","declaration":"theorem sum_eq_int_deriv_aux1 {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_le_kpOne : b ≤ ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} | |
| {"name":"integrability_aux","declaration":"theorem integrability_aux {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => ↑⌊x⌋ + 1 / 2 - ↑x) MeasureTheory.volume a b"} | |
| {"name":"deriv_fun_re","declaration":"theorem deriv_fun_re {t : ℝ} {f : ℂ → ℂ} (diff : ∀ (σ : ℝ), DifferentiableAt ℂ f (↑σ + ↑t * Complex.I)) : (deriv fun {σ₂} => f (↑σ₂ + ↑t * Complex.I)) = fun σ => deriv f (↑σ + ↑t * Complex.I)"} | |
| {"name":"ZetaSum_aux1₁","declaration":"theorem ZetaSum_aux1₁ {a : ℕ} {b : ℕ} {s : ℂ} (s_ne_one : s ≠ 1) (ha : a ∈ Set.Ioo 0 b) : ∫ (x : ℝ) in ↑a..↑b, 1 / ↑x ^ s = (↑b ^ (1 - s) - ↑a ^ (1 - s)) / (1 - s)"} | |
| {"name":"ZetaUpperBnd","declaration":"theorem ZetaUpperBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n ↑ct_aux1 < |t| → σ ∈ Set.Icc (1 - A / Real.log |t|) 2 → ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t|"} | |
| {"name":"Zeta_eq_int_derivZeta","declaration":"theorem Zeta_eq_int_derivZeta {σ₁ : ℝ} {σ₂ : ℝ} {t : ℝ} (t_ne_zero : t ≠ 0) : ∫ (σ : ℝ) in σ₁..σ₂, deriv riemannZeta (↑σ + ↑t * Complex.I) =\n riemannZeta (↑σ₂ + ↑t * Complex.I) - riemannZeta (↑σ₁ + ↑t * Complex.I)"} | |
| {"name":"ZetaSum_aux2","declaration":"theorem ZetaSum_aux2 {N : ℕ} (N_pos : 0 < N) {s : ℂ} (s_re_gt : 1 < s.re) : ∑' (n : ℕ), 1 / (↑n + ↑N) ^ s =\n -↑N ^ (1 - s) / (1 - s) - ↑N ^ (-s) / 2 + s * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1))"} | |
| {"name":"Complex.cpow_tendsto","declaration":"theorem Complex.cpow_tendsto {s : ℂ} (s_re_gt : 1 < s.re) : Filter.Tendsto (fun x => ↑x ^ (1 - s)) Filter.atTop (nhds 0)"} | |
| {"name":"ZetaSum_aux1_4'","declaration":"theorem ZetaSum_aux1_4' (x : ℝ) (hx : 0 < x) (s : ℂ) : ‖(↑⌊x⌋ + 1 / 2 - ↑x) / ↑x ^ (s + 1)‖ = |↑⌊x⌋ + 1 / 2 - x| / x ^ (s + 1).re"} | |
| {"name":"uIcc_subsets","declaration":"theorem uIcc_subsets {a : ℝ} {b : ℝ} {c : ℝ} (hc : c ∈ Set.Icc a b) : Set.uIcc a c ⊆ Set.uIcc a b ∧ Set.uIcc c b ⊆ Set.uIcc a b"} | |
| {"name":"Real.differentiableAt_cpow_const_of_ne","declaration":"theorem Real.differentiableAt_cpow_const_of_ne (s : ℂ) {x : ℝ} (xpos : 0 < x) : DifferentiableAt ℝ (fun x => ↑x ^ s) x"} | |
| {"name":"neg_s_ne_neg_one","declaration":"theorem neg_s_ne_neg_one {s : ℂ} (s_ne_one : s ≠ 1) : -s ≠ -1"} | |
| {"name":"UpperBnd_aux6","declaration":"theorem UpperBnd_aux6 {σ : ℝ} {t : ℝ} (t_ge : 3 < |t|) (σ_gt : 1 / 2 < σ) (σ_le : σ ≤ 2) (neOne : ↑σ + ↑t * Complex.I ≠ 1) (Npos : 0 < ⌊|t|⌋₊) (N_le_t : ↑⌊|t|⌋₊ ≤ |t|) : ↑⌊|t|⌋₊ ^ (1 - σ) / ‖1 - (↑σ + ↑t * Complex.I)‖ ≤ |t| ^ (1 - σ) * 2 ∧\n ↑⌊|t|⌋₊ ^ (-σ) / 2 ≤ |t| ^ (1 - σ) ∧ ↑⌊|t|⌋₊ ^ (-σ) / σ ≤ 8 * |t| ^ (-σ)"} | |
| {"name":"norm_add₄_le","declaration":"theorem norm_add₄_le {E : Type u_1} [SeminormedAddGroup E] (a : E) (b : E) (c : E) (d : E) : ‖a + b + c + d‖ ≤ ‖a‖ + ‖b‖ + ‖c‖ + ‖d‖"} | |
| {"name":"ZetaSum_aux1_2","declaration":"theorem ZetaSum_aux1_2 {a : ℝ} {b : ℝ} {c : ℝ} (apos : 0 < a) (a_lt_b : a < b) (h : c ≠ 0 ∧ 0 ∉ Set.uIcc a b) : ∫ (x : ℝ) in a..b, 1 / x ^ (c + 1) = (a ^ (-c) - b ^ (-c)) / c"} | |
| {"name":"sum_eq_int_deriv_aux_lt","declaration":"theorem sum_eq_int_deriv_aux_lt {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (ha : a ∈ Set.Ico (↑k) b) (b_lt_kpOne : b < ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} | |
| {"name":"ZetaSum_aux1_5a","declaration":"theorem ZetaSum_aux1_5a {a : ℝ} {b : ℝ} (apos : 0 < a) {s : ℂ} (x : ℝ) (h : x ∈ Set.Icc a b) : |↑⌊x⌋ + 1 / 2 - x| / x ^ (s.re + 1) ≤ 1 / x ^ (s.re + 1)"} | |
| {"name":"le_trans₄","declaration":"theorem le_trans₄ {α : Type u_1} [Preorder α] {a : α} {b : α} {c : α} {d : α} : a ≤ b → b ≤ c → c ≤ d → a ≤ d"} | |
| {"name":"integrability_aux₂","declaration":"theorem integrability_aux₂ {a : ℝ} {b : ℝ} : IntervalIntegrable (fun x => 1 / 2 - ↑x) MeasureTheory.volume a b"} | |
| {"name":"ZetaNear1BndFilter","declaration":"theorem ZetaNear1BndFilter : (fun σ => riemannZeta ↑σ) =O[nhdsWithin 1 (Set.Ioi 1)] fun σ => 1 / (↑σ - 1)"} | |
| {"name":"ZetaNear1BndExact","declaration":"theorem ZetaNear1BndExact : ∃ c, ∃ (_ : 0 < c), ∀ σ ∈ Set.Ioc 1 2, ‖riemannZeta ↑σ‖ ≤ c / (σ - 1)"} | |
| {"name":"LogDerivZetaBnd","declaration":"theorem LogDerivZetaBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Ico (1 - A / Real.log |t| ^ 9) 1 →\n ‖deriv riemannZeta (↑σ + ↑t * Complex.I) / riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 9"} | |
| {"name":"ZetaSum_aux1_3a","declaration":"theorem ZetaSum_aux1_3a (x : ℝ) : -(1 / 2) < ↑⌊x⌋ + 1 / 2 - x"} | |
| {"name":"UpperBnd_aux","declaration":"theorem UpperBnd_aux {A : ℝ} {σ : ℝ} {t : ℝ} (A_pos : 0 < A) (A_lt : A < 1) (t_ge : 3 < |t|) (σ_ge : 1 - A / Real.log |t| ≤ σ) : 1 < Real.log |t| ∧ 1 - A < σ ∧ 0 < σ ∧ ↑σ + ↑t * Complex.I ≠ 1"} | |
| {"name":"ZetaSum_aux1φDiff","declaration":"theorem ZetaSum_aux1φDiff {s : ℂ} {x : ℝ} (xpos : 0 < x) : HasDerivAt (fun t => 1 / ↑t ^ s) (deriv (fun t => 1 / ↑t ^ s) x) x"} | |
| {"name":"Complex.one_div_cpow_eq","declaration":"theorem Complex.one_div_cpow_eq {s : ℂ} {x : ℝ} (x_ne : x ≠ 0) : 1 / ↑x ^ s = ↑x ^ (-s)"} | |
| {"name":"ZetaInvBnd","declaration":"theorem ZetaInvBnd : ∃ A,\n ∃ (_ : 0 < A),\n ∃ C,\n ∃ (_ : 0 < C),\n ∀ (σ t : ℝ),\n 3 < |t| →\n σ ∈ Set.Ico (1 - A / Real.log |t| ^ 9) 1 → 1 / ‖riemannZeta (↑σ + ↑t * Complex.I)‖ ≤ C * Real.log |t| ^ 7"} | |
| {"name":"riemannZeta0_apply","declaration":"theorem riemannZeta0_apply (N : ℕ) (s : ℂ) : riemannZeta0 N s =\n (Finset.sum (Finset.range N) fun n => 1 / ↑n ^ s) +\n (-↑N ^ (1 - s) / (1 - s) + -↑N ^ (-s) / 2 + s * ∫ (x : ℝ) in Set.Ioi ↑N, (↑⌊x⌋ + 1 / 2 - ↑x) * ↑x ^ (-(s + 1)))"} | |
| {"name":"div_rpow_eq_rpow_neg","declaration":"theorem div_rpow_eq_rpow_neg (a : ℝ) (x : ℝ) (s : ℝ) (hx : 0 ≤ x) : a / x ^ s = a * x ^ (-s)"} | |
| {"name":"HolomophicOn_riemannZeta","declaration":"theorem HolomophicOn_riemannZeta : HolomorphicOn riemannZeta {s | s ≠ 1}"} | |
| {"name":"sum_eq_int_deriv_aux2","declaration":"theorem sum_eq_int_deriv_aux2 {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} (c : ℂ) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : ∫ (x : ℝ) in a..b, (c - ↑x) * deriv φ x = (c - ↑b) * φ b - (c - ↑a) * φ a + ∫ (x : ℝ) in a..b, φ x"} | |
| {"name":"riemannZeta0","declaration":"def riemannZeta0 (N : ℕ) (s : ℂ) : ℂ"} | |
| {"name":"ZetaSum_aux1derivφCont","declaration":"theorem ZetaSum_aux1derivφCont {s : ℂ} (s_ne_zero : s ≠ 0) {a : ℕ} {b : ℕ} (ha : a ∈ Set.Ioo 0 b) : ContinuousOn (deriv fun t => 1 / ↑t ^ s) (Set.uIcc ↑a ↑b)"} | |
| {"name":"sum_eq_int_deriv_aux_eq","declaration":"theorem sum_eq_int_deriv_aux_eq {φ : ℝ → ℂ} {a : ℝ} {b : ℝ} {k : ℤ} (b_eq_kpOne : b = ↑k + 1) (φDiff : ∀ x ∈ Set.uIcc a b, HasDerivAt φ (deriv φ x) x) (derivφCont : ContinuousOn (deriv φ) (Set.uIcc a b)) : (Finset.sum (Finset.Ioc k ⌊b⌋) fun n => φ ↑n) =\n (∫ (x : ℝ) in a..b, φ x) + (↑⌊b⌋ + 1 / 2 - ↑b) * φ b - (↑k + 1 / 2 - ↑a) * φ a -\n ∫ (x : ℝ) in a..b, (↑k + 1 / 2 - ↑x) * deriv φ x"} | |
| {"name":"Finset.sum_Ioc_add_sum_Ioc","declaration":"theorem Finset.sum_Ioc_add_sum_Ioc {a : ℤ} {b : ℤ} {c : ℤ} (f : ℤ → ℂ) (hb : b ∈ Finset.Icc a c) : ((Finset.sum (Finset.Ioc a b) fun n => f n) + Finset.sum (Finset.Ioc b c) fun n => f n) =\n Finset.sum (Finset.Ioc a c) fun n => f n"} | |
| {"name":"ZetaSum_aux2a","declaration":"theorem ZetaSum_aux2a : ∃ C, ∀ (x : ℝ), |↑⌊x⌋ + 1 / 2 - x| ≤ C"} | |