Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Neg DE₂ term re pos

negDE₂_term_re_pos

sphere packing Fourier analysis modular forms fg

Plain-language statement

Each term n*σ₁(n)*exp(-2πnt) in the q-expansion of negDE₂ has positive real part.

Exact Lean statement

lemma negDE₂_term_re_pos (t : ℝ) (ht : 0 < t) (n : ℕ+) :
    0 < ((n : ℂ) * (ArithmeticFunction.sigma 1 n : ℂ) *
      Complex.exp (2 * ↑Real.pi * Complex.I * ↑n *
        ↑(⟨Complex.I * t, by simp [ht]⟩ : UpperHalfPlane))).re

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Lean source

Canonical source

Full Lean sourceLean 4

lemma negDE₂_term_re_pos (t : ℝ) (ht : 0 < t) (n : ℕ+) :    0 < ((n : ℂ) * (ArithmeticFunction.sigma 1 n : ℂ) *      Complex.exp (2 * ↑Real.pi * Complex.I * ↑n *        ↑(⟨Complex.I * t, by simp [ht]⟩ : UpperHalfPlane))).re := by  rw [qexp_arg_imag_axis_pnat t ht n]  simp only [Complex.mul_re, Complex.exp_ofReal_re, Complex.exp_ofReal_im, mul_zero,    sub_zero, Complex.natCast_re, Complex.natCast_im]  refine mul_pos (mul_pos ?_ ?_) (Real.exp_pos _)  · exact_mod_cast n.pos  · exact_mod_cast ArithmeticFunction.sigma_pos 1 n n.ne_zero

Project: Sphere Packing in Dimension 8
License: Apache-2.0
Commit: acfc6204e65a
Source: SpherePacking/ModularForms/FG.lean:511-520

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Project-declaredLean 4.31.0

Anti Der Pos

antiDerPos

Plain-language statement

If $F$ is a modular form where $F(it)$ is positive for sufficiently large $t$ (i.e. constant term is positive) and the derivative is positive, then $F$ is also positive.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

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Project-declaredLean 4.31.0

Let $F : \mathbb{H} \to \mathbb{C}$ be a holomorphic function where $F(it)$ is real for all $t > 0$ . Assume that Serre derivative $\partial_k F$ is positive on the imaginary axis. If $F(it)$ is positive for sufficiently large $t$ , then $F(it)$ is positive for all $t > 0$ .

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Project-declaredLean 4.31.0

Closed Ball center subset upper Half Plane

closedBall_center_subset_upperHalfPlane

Plain-language statement

Closed ball centered at z with radius z.im/2 is contained in the upper half plane.

sphere packingFourier analysismodular forms

Source project: Sphere Packing in Dimension 8

Person-level attribution pending.

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Anti Der Pos

Anti Serre Der Pos

Closed Ball center subset upper Half Plane