Theorem A from three hypotheses (explicit formula, Riemann–von Mangoldt, $\Gamma$ -facts)

Proved

Zeta23.thmA3

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

riemann-zetazero-countingzeta23

Here $N(T, 2T)$ (Ncount) counts the nontrivial zeros $\rho$ of Mathlib's riemannZeta ( $\zeta(\rho) = 0$ , $0 < \operatorname{Re}\rho < 1$ ) with $T < \operatorname{Im}\rho \le 2T$ , with multiplicity, and $N_0^*(T, 2T)$ (N0star) counts the distinct such zeros on the critical line. The conclusion is Theorem A's dyadic $\varepsilon$ -form: for every $\varepsilon > 0$ there is $T_0$ such that for all $T \ge T_0$ ,

\Big(\frac{2}{3} - \varepsilon\Big)\, N(T, 2T) \;\le\; N_0^*(T, 2T).

As stated in the repository (Zeta23/Final.lean, section ThreeHyp), this version carries three hypotheses as section variables: hEF : EF.EF_lit zetaZeroConfig (Weil's explicit formula in literature form [eq:EFstd]), hRvM : RiemannVonMangoldt zetaZeroConfig (the Riemann–von Mangoldt asymptotic [eq:RvM] with local count), and hΓ : GammaFacts (Stirling-type facts for the density $\mu$ , [eq:mufacts]/[eq:muints]); the uploaded standalone statement displays only the conclusion. The Chebyshev–Mertens bounds and the Montgomery–Vaughan inequality, hypotheses of the five-input version, are here already discharged as theorems of the repository.

It is consumed by Zeta23.thmA1, which removes the RvM and $\Gamma$ hypotheses (both proved unconditionally in the repository), leaving the explicit formula as the single remaining assumption.

Preamble

import Batteries.Tactic.Lemma
import Mathlib
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Field.GeomSum
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.Algebra.Order.Star.Basic
import Mathlib.Analysis.Analytic.Order
import Mathlib.Analysis.Analytic.Uniqueness
import Mathlib.Analysis.Asymptotics.Lemmas
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.Fourier.PoissonSummation
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Matrix.Normed
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.ExpDeriv
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.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.SmoothTransition
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.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
import Mathlib.MeasureTheory.Order.Group.Lattice
import Mathlib.NumberTheory.AbelSummation
import Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.Harmonic.Bounds
import Mathlib.NumberTheory.Harmonic.EulerMascheroni
import Mathlib.NumberTheory.Harmonic.GammaDeriv
import Mathlib.NumberTheory.Harmonic.ZetaAsymp
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.Order
import Mathlib.Topology.Algebra.InfiniteSum.Real
import Mathlib.Topology.ContinuousMap.Bounded.Basic
import Mathlib.Topology.Instances.Matrix
import Definitions.Def_Zeta23_Defs
import Definitions.Def_Zeta23_ExplicitFormula
import Definitions.Def_Zeta23_Hypotheses
import Definitions.Def_Zeta23_MV
import Definitions.Def_Zeta23_Statement
import Definitions.Def_Zeta23_Statement_Seam
import Definitions.Def_Zeta23_Statement_SeamClosed
import Definitions.Def_Zeta23_ZetaReflect

open Filter
open Zeta23
variable (hEF : EF.EF_lit zetaZeroConfig) (hRvM : RiemannVonMangoldt zetaZeroConfig) (hΓ : GammaFacts)
include hEF hRvM hΓ

Formal statement

theorem Zeta23.thmA3 :
    ∀ ε > 0, ∃ T₀ : ℝ, ∀ T ≥ T₀, (2 / 3 - ε) * (Ncount T (2 * T) : ℝ) ≤ N0star T (2 * T) := by sorry

Source

https://github.com/anthropics/zeta-23-lean/blob/182afbf851aa42a8ae78507be83f2356d3a33260/Zeta23/Final.lean#L140-L145

View graph

Theorem A from three hypotheses (explicit formula, Riemann–von Mangoldt, Γ\GammaΓ-facts)

Theorem A from three hypotheses (explicit formula, Riemann–von Mangoldt, $\Gamma$ -facts)