Dyadic-to-cumulative wrapper for zero-count lower bounds

Proved

Zeta23.cumulative_of_dyadic

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

riemann-zetazero-countingzeta23

Here Ncount $= N(T_1, T_2)$ counts the nontrivial zeros $\rho$ of Mathlib's riemannZeta ( $\zeta(\rho) = 0$ , $0 < \operatorname{Re}\rho < 1$ ) with $T_1 < \operatorname{Im}\rho \le T_2$ , with multiplicity; the ambient section variable hs : ZetaSeam packages the classical $\zeta$ -facts needed to view these zeros as an abstract zero configuration zetaZeros hs, and the hypothesis hR is the Riemann–von Mangoldt property [eq:RvM] for that configuration.

Let $c \in \mathbb{R}$ and let $f : \mathbb{R} \to \mathbb{R} \to \mathbb{N}$ be any interval-additive two-parameter count: $f(a,d) = f(a,b) + f(b,d)$ whenever $a \le b \le d$ . Assume the dyadic $\varepsilon$ -form against $N$ : for every $\varepsilon > 0$ there is $T_0$ such that for all $T \ge T_0$ ,

(c - \varepsilon)\, N(T, 2T) \;\le\; f(T, 2T).

Then the cumulative $\varepsilon$ -form holds: for every $\varepsilon > 0$ there is $T_0$ such that for all $T \ge T_0$ ,

(c - \varepsilon)\, N(0, T) \;\le\; f(0, T).

This is the generic wrapper (in Zeta23.Main) that sums a lower bound over dyadic windows $(T, 2T]$ into a bound on the full window $(0, T]$ , using Riemann–von Mangoldt to control the growth of $N$ ; instantiated with $f = N_0^*$ and $c = 2/3$ it turns Theorem A's dyadic form into its cumulative form. It is consumed by Zeta23.thmA_cumulative_of_traces.

Preamble

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.Order.Chebyshev
import Mathlib.Algebra.Order.Field.GeomSum
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.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Complex.ReImTopology
import Mathlib.Analysis.Convex.Birkhoff
import Mathlib.Analysis.Fourier.Convolution
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.Fourier.Inversion
import Mathlib.Analysis.Fourier.PoissonSummation
import Mathlib.Analysis.Matrix.Normed
import Mathlib.Analysis.Matrix.PosDef
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.Module.Connected
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.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.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.Set.Card
import Mathlib.LinearAlgebra.Complex.FiniteDimensional
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
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.Prod
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.AbelSummation
import Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt
import Mathlib.NumberTheory.Chebyshev
import Mathlib.NumberTheory.Harmonic.EulerMascheroni
import Mathlib.NumberTheory.Harmonic.GammaDeriv
import Mathlib.NumberTheory.LSeries.Nonvanishing
import Mathlib.NumberTheory.LSeries.RiemannZeta
import Mathlib.Order.Filter.AtTopBot.Field
import Mathlib.Topology.Algebra.InfiniteSum.Order
import Mathlib.Topology.Algebra.InfiniteSum.Real
import Mathlib.Topology.Instances.Matrix
import Definitions.Def_Zeta23_Assembly
import Definitions.Def_Zeta23_Assembly_Inputs
import Definitions.Def_Zeta23_Defs
import Definitions.Def_Zeta23_Defs_Profile
import Definitions.Def_Zeta23_ExplicitFormula
import Definitions.Def_Zeta23_FromPNTPlus_EulerMaclaurin
import Definitions.Def_Zeta23_FromPNTPlus_Mertens
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_Main
import Definitions.Def_Zeta23_Poisson
import Definitions.Def_Zeta23_PrimeSideTemp
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_Taper_Basic
import Definitions.Def_Zeta23_Taper_Params
import Definitions.Def_Zeta23_TracesBoundsE
import Definitions.Def_Zeta23_ZeroSide
import Definitions.Def_Zeta23_ZetaReflect

open Filter Topology
open Zeta23
variable (hs : ZetaSeam)

Formal statement

theorem Zeta23.cumulative_of_dyadic (hR : RiemannVonMangoldt (zetaZeros hs)) {c : ℝ} {f : ℝ → ℝ → ℕ}
    (hf_add : ∀ a b d : ℝ, a ≤ b → b ≤ d → f a d = f a b + f b d)
    (h : ∀ ε > 0, ∃ T₀ : ℝ, ∀ T ≥ T₀, (c - ε) * (Ncount T (2 * T) : ℝ) ≤ f T (2 * T)) :
    ∀ ε > 0, ∃ T₀ : ℝ, ∀ T ≥ T₀, (c - ε) * (Ncount 0 T : ℝ) ≤ f 0 T := by sorry

Source

https://github.com/anthropics/zeta-23-lean/blob/182afbf851aa42a8ae78507be83f2356d3a33260/Zeta23/Main.lean#L49-L65

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