File: Mathlib/NumberTheory/LSeries/Convergence.lean

full_name: LSeries.abscissaOfAbsConv
code: /-- The abscissa `x : EReal` of absolute convergence of the L-series associated to `f`:
the series converges absolutely at `s` when `re s > x` and does not converge absolutely
when `re s < x`. -/
noncomputable def LSeries.abscissaOfAbsConv (f : ℕ → ℂ) : EReal 
goal: EReal

full_name: LSeries.abscissaOfAbsConv_congr
code: lemma LSeries.abscissaOfAbsConv_congr {f g : ℕ → ℂ} (h : ∀ {n}, n ≠ 0 → f n = g n) :
    abscissaOfAbsConv f = abscissaOfAbsConv g 
goal: LSeries.abscissaOfAbsConv f = LSeries.abscissaOfAbsConv g

full_name: LSeries.abscissaOfAbsConv_congr'
code: /-- If `f` and `g` agree on large `n : ℕ`, then their `LSeries` have the same
abscissa of absolute convergence. -/
lemma LSeries.abscissaOfAbsConv_congr' {f g : ℕ → ℂ} (h : f =ᶠ[atTop] g) :
    abscissaOfAbsConv f = abscissaOfAbsConv g 
goal: LSeries.abscissaOfAbsConv f = LSeries.abscissaOfAbsConv g

full_name: LSeriesSummable_of_abscissaOfAbsConv_lt_re
code: lemma LSeriesSummable_of_abscissaOfAbsConv_lt_re {f : ℕ → ℂ} {s : ℂ}
    (hs : abscissaOfAbsConv f < s.re) : LSeriesSummable f s 
goal: LSeriesSummable f s

full_name: LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re
code: lemma LSeriesSummable_lt_re_of_abscissaOfAbsConv_lt_re {f : ℕ → ℂ} {s : ℂ}
    (hs : abscissaOfAbsConv f < s.re) :
    ∃ x : ℝ, x < s.re ∧ LSeriesSummable f x 
goal: ∃ x < s.re, LSeriesSummable f ↑x

full_name: LSeriesSummable.abscissaOfAbsConv_le
code: lemma LSeriesSummable.abscissaOfAbsConv_le {f : ℕ → ℂ} {s : ℂ} (h : LSeriesSummable f s) :
    abscissaOfAbsConv f ≤ s.re 
goal: LSeries.abscissaOfAbsConv f ≤ ↑s.re

full_name: LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable
code: lemma LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable {f : ℕ → ℂ} {x : ℝ}
    (h : ∀ y : ℝ, x < y → LSeriesSummable f y) :
    abscissaOfAbsConv f ≤ x 
goal: LSeries.abscissaOfAbsConv f ≤ ↑x

full_name: LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable'
code: lemma LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable' {f : ℕ → ℂ} {x : EReal}
    (h : ∀ y : ℝ, x < y → LSeriesSummable f y) :
    abscissaOfAbsConv f ≤ x 
goal: LSeries.abscissaOfAbsConv f ≤ x

full_name: LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow
code: /-- If `‖f n‖` is bounded by a constant times `n^x`, then the abscissa of absolute convergence
of `f` is bounded by `x + 1`. -/
lemma LSeries.abscissaOfAbsConv_le_of_le_const_mul_rpow {f : ℕ → ℂ} {x : ℝ}
    (h : ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C * n ^ x) : abscissaOfAbsConv f ≤ x + 1 
goal: LSeries.abscissaOfAbsConv f ≤ ↑x + 1

full_name: LSeries.abscissaOfAbsConv_le_of_isBigO_rpow
code: /-- If `‖f n‖` is `O(n^x)`, then the abscissa of absolute convergence
of `f` is bounded by `x + 1`. -/
lemma LSeries.abscissaOfAbsConv_le_of_isBigO_rpow {f : ℕ → ℂ} {x : ℝ}
    (h : f =O[atTop] fun n ↦ (n : ℝ) ^ x) :
    abscissaOfAbsConv f ≤ x + 1 
goal: LSeries.abscissaOfAbsConv f ≤ ↑x + 1

full_name: LSeries.abscissaOfAbsConv_le_of_le_const
code: /-- If `f` is bounded, then the abscissa of absolute convergence of `f` is bounded above by `1`. -/
lemma LSeries.abscissaOfAbsConv_le_of_le_const {f : ℕ → ℂ} (h : ∃ C, ∀ n ≠ 0, ‖f n‖ ≤ C) :
    abscissaOfAbsConv f ≤ 1 
goal: LSeries.abscissaOfAbsConv f ≤ 1

full_name: LSeries.abscissaOfAbsConv_le_one_of_isBigO_one
code: /-- If `f` is `O(1)`, then the abscissa of absolute convergence of `f` is bounded above by `1`. -/
lemma LSeries.abscissaOfAbsConv_le_one_of_isBigO_one {f : ℕ → ℂ} (h : f =O[atTop] (1 : ℕ → ℝ)) :
    abscissaOfAbsConv f ≤ 1 
goal: LSeries.abscissaOfAbsConv f ≤ 1