DE₄ imag axis re pos
DE₄_imag_axis_re_pos
Plain-language statement
The real part of (D E₄)(it) is positive for t > 0.
Exact Lean statement
lemma DE₄_imag_axis_re_pos (t : ℝ) (ht : 0 < t) :
0 < ((D E₄.toFun).resToImagAxis t).reFormal artifact
Lean source
lemma DE₄_imag_axis_re_pos (t : ℝ) (ht : 0 < t) : 0 < ((D E₄.toFun).resToImagAxis t).re := by simp only [Function.resToImagAxis, ResToImagAxis, ht, ↓reduceDIte] set z : UpperHalfPlane := ⟨Complex.I * t, by simp [ht]⟩ with hz rw [DE₄_qexp z] have hsum : Summable fun n : ℕ+ => (n : ℂ) * (ArithmeticFunction.sigma 3 n : ℂ) * Complex.exp (2 * ↑Real.pi * Complex.I * n * z) := by simp only [hz]; exact DE₄_summable t ht have hsum_re : Summable fun n : ℕ+ => ((n : ℂ) * (ArithmeticFunction.sigma 3 n : ℂ) * Complex.exp (2 * ↑Real.pi * Complex.I * n * z)).re := ⟨_, Complex.hasSum_re hsum.hasSum⟩ have hpos : ∀ n : ℕ+, 0 < ((n : ℂ) * (ArithmeticFunction.sigma 3 n : ℂ) * Complex.exp (2 * ↑Real.pi * Complex.I * n * z)).re := by intro n simp only [hz] exact DE₄_term_re_pos t ht n have htsum_pos := Summable.tsum_pos hsum_re (fun n => (hpos n).le) 1 (hpos 1) simp only [Complex.mul_re, Complex.re_ofNat, Complex.im_ofNat, zero_mul, sub_zero] rw [Complex.re_tsum hsum] exact mul_pos (by norm_num : (0 : ℝ) < 240) htsum_pos- Project
- Sphere Packing in Dimension 8
- License
- Apache-2.0
- Commit
- acfc6204e65a
- Source
- SpherePacking/ModularForms/FG.lean:440-459
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Related declarations
Anti Der Pos
antiDerPos
Plain-language statement
If is a modular form where is positive for sufficiently large (i.e. constant term is positive) and the derivative is positive, then is also positive.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
Anti Serre Der Pos
antiSerreDerPos
Plain-language statement
Let be a holomorphic function where is real for all . Assume that Serre derivative is positive on the imaginary axis. If is positive for sufficiently large , then is positive for all .
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.
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closedBall_center_subset_upperHalfPlane
Plain-language statement
Closed ball centered at z with radius z.im/2 is contained in the upper half plane.
Source project: Sphere Packing in Dimension 8
Person-level attribution pending.