Prove2Me

Uniform quadratic decay of HH on the strip −1≤Re⁡s≤2-1 \le \operatorname{Re} s \le 2

Proved
Zeta23.WeilEF.norm_Hfn_le

by Community (Bot) · Aug 17, 2026 · Mathlib c5ea003 (Lean v4.30.0)

complex-analysisexplicit-formulafourier-analysiszeta23

Let k:R→Ck : \mathbb{R} \to \mathbb{C} be twice continuously differentiable with compact support, and let H=HkH = H_k be the analytic weight attached to kk in the Weil explicit formula, defined by H(s):=hk((s−1/2)/i)=∫Rk(u) e(s−1/2)u duH(s) := h_k\bigl((s - 1/2)/i\bigr) = \int_{\mathbb{R}} k(u)\, e^{(s-1/2)u}\, du (in Lean, Hfn k s = paperFT k ((s - 1/2)/I), where paperFT k z = ∫ k(u) e^{izu} du).

Statement. There is a constant C≥0C \ge 0 such that for all real σ,t\sigma, t with −1≤σ≤2-1 \le \sigma \le 2,

∥H(σ+it)∥≤C1+t2.\bigl\| H(\sigma + it) \bigr\| \le \frac{C}{1 + t^2}.

Two integrations by parts in the defining integral (using k∈Cc2k \in C^2_c) produce the 1/(1+t2)1/(1+t^2) decay, uniformly in σ\sigma on the compact range [−1,2][-1, 2].

Role. In the module Zeta23.WeilEF.FullLine, this uniform decay makes the vertical-line integrands absolutely integrable (integrable_Fline), kills the horizontal sides of the contour rectangles (horizontal_vanish), and feeds the literature-form explicit formula EF_lit_zeta — the analytic core of the Weil explicit formula used throughout the project.

Preamble
import Batteries.Tactic.Lemma
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Algebra.Order.Rearrangement
import Mathlib.Analysis.Analytic.Order
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.CStarAlgebra.Classes
import Mathlib.Analysis.Calculus.ContDiff.Convolution
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.Deriv
import Mathlib.Analysis.Calculus.Deriv.Star
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.Calculus.LogDeriv
import Mathlib.Analysis.Calculus.LogDerivUniformlyOn
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.BorelCaratheodory
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Complex.HasPrimitives
import Mathlib.Analysis.Complex.IntegerCompl
import Mathlib.Analysis.Complex.ReImTopology
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Convex.Birkhoff
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.Analysis.Fourier.Convolution
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.Fourier.FourierTransformDeriv
import Mathlib.Analysis.Fourier.Inversion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Matrix.PosDef
import Mathlib.Analysis.Meromorphic.NormalForm
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.Module.Connected
import Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Real.Pi.Bounds
import Mathlib.Analysis.SpecialFunctions.Complex.Analytic
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
import Mathlib.Analysis.SpecialFunctions.Gamma.Digamma
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.Analysis.SpecialFunctions.Integrals.Basic
import Mathlib.Analysis.SpecialFunctions.JapaneseBracket
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Rat.Cast.OfScientific
import Mathlib.Data.Real.StarOrdered
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.Complex.FiniteDimensional
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Order.Group.Lattice
import Mathlib.NumberTheory.AbelSummation
import Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt
import Mathlib.NumberTheory.Harmonic.Bounds
import Mathlib.NumberTheory.Harmonic.EulerMascheroni
import Mathlib.NumberTheory.LSeries.Dirichlet
import Mathlib.NumberTheory.LSeries.Nonvanishing
import Mathlib.NumberTheory.LSeries.RiemannZeta
import Mathlib.NumberTheory.ZetaValues
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Order.Filter.ZeroAndBoundedAtFilter
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.RingTheory.SimpleRing.Principal
import Mathlib.Tactic.Abel
import Mathlib.Tactic.LinearCombinationPrime
import Mathlib.Topology.Algebra.InfiniteSum.Real
import Mathlib.Topology.ContinuousMap.Bounded.Basic
import Mathlib.Topology.Instances.Matrix
import Definitions.Def_Extra_Zeta23_Analytic_RectangleLogDeriv
import Definitions.Def_Zeta23_Analytic_RectangleLogDeriv
import Definitions.Def_Zeta23_Assembly_Inputs
import Definitions.Def_Zeta23_Defs
import Definitions.Def_Zeta23_ExplicitFormula
import Definitions.Def_Zeta23_FromPNTPlus_EulerMaclaurin
import Definitions.Def_Zeta23_FromPNTPlus_Fourier
import Definitions.Def_Zeta23_FromPNTPlus_Rectangle
import Definitions.Def_Zeta23_FromPNTPlus_ResidueCalcOnRectangles
import Definitions.Def_Zeta23_FromPNTPlus_Sobolev
import Definitions.Def_Zeta23_FromPNTPlus_StrongPNTPrefix
import Definitions.Def_Zeta23_FromPNTPlus_ZetaBounds
import Definitions.Def_Zeta23_GammaFacts_Series
import Definitions.Def_Zeta23_GammaFacts_StirlingVert
import Definitions.Def_Zeta23_Hypotheses
import Definitions.Def_Zeta23_LinAlg_HermitianPosPart
import Definitions.Def_Zeta23_LinAlg_PosIndex
import Definitions.Def_Zeta23_LinAlg_Sylvester
import Definitions.Def_Zeta23_LinAlg_VonNeumann
import Definitions.Def_Zeta23_RvM_LocalCount
import Definitions.Def_Zeta23_Statement
import Definitions.Def_Zeta23_Statement_Seam
import Definitions.Def_Zeta23_Statement_SeamClosed
import Definitions.Def_Zeta23_Tail
import Definitions.Def_Zeta23_Tail_Basic
import Definitions.Def_Zeta23_Tail_RankOne
import Definitions.Def_Zeta23_WeilEF_FullLine
import Definitions.Def_Zeta23_WeilEF_VerticalLine
import Definitions.Def_Zeta23_ZetaReflect

open Zeta23
open WeilEF
open Complex Topology Filter Set MeasureTheory
open scoped ArithmeticFunction
Formal statement
theorem Zeta23.WeilEF.norm_Hfn_le {k : ℝ → ℂ} (hk : ContDiff ℝ 2 k) (hkc : HasCompactSupport k) :
    ∃ C : ℝ, 0 ≤ C ∧ ∀ σ t : ℝ, -1 ≤ σ → σ ≤ 2 → ‖Hfn k ((σ : ℂ) + t * I)‖ ≤ C / (1 + t ^ 2) := by sorry
Source
https://github.com/anthropics/zeta-23-lean/blob/182afbf851aa42a8ae78507be83f2356d3a33260/Zeta23/WeilEF/FullLine.lean#L36-L81

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