Interval additivity of $N_0^*$

Proved

Zeta23.Assembly.N0star_add

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

number-theoryzero-countingzeta23

For an abstract zero configuration $Z$ (a locally finite set of points $\rho = \beta + i\gamma$ in the strip $0 \le \beta \le 1$ with multiplicities, invariant under $\rho \mapsto 1 - \bar\rho$ ), $N_0^*(T_1, T_2)$ denotes the number of distinct zeros on the critical line $\beta = 1/2$ with ordinate in the half-open window $T_1 < \gamma \le T_2$ (counted without multiplicity).

The theorem asserts that this count is additive over adjacent windows: for real numbers $a \le b \le c$ ,

N_0^*(a, c) = N_0^*(a, b) + N_0^*(b, c).

The half-open convention $T_1 < \gamma \le T_2$ is what makes the two windows disjoint and the identity exact.

This bookkeeping fact underlies both the seam inequality seamA (assembling the zero-side bound on $N_0^*(T, 2T)$ ) and the dyadic-summation passage from the dyadic statement $N_0^*(T,2T) \ge (2/3-\varepsilon)N(T,2T)$ to the cumulative form of Theorem A (thmA_cumulative_of_traces).

Preamble

import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Chebyshev
import Mathlib.Algebra.Order.Rearrangement
import Mathlib.Analysis.CStarAlgebra.Classes
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Complex.ExponentialBounds
import Mathlib.Analysis.Convex.Birkhoff
import Mathlib.Analysis.Matrix.Normed
import Mathlib.Analysis.Matrix.PosDef
import Mathlib.Analysis.Real.Pi.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Digamma
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Set.Card
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt
import Definitions.Def_Zeta23_Assembly
import Definitions.Def_Zeta23_Assembly_Inputs
import Definitions.Def_Zeta23_Defs
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_PrimeSideTemp
import Definitions.Def_Zeta23_TracesBoundsE

open Matrix Finset RHLinalg
open scoped ComplexOrder
open Zeta23
open Assembly
open Set
variable (Z : ZeroConfig)

Formal statement

theorem Zeta23.Assembly.N0star_add {a b c : ℝ} (hab : a ≤ b) (hbc : b ≤ c) :
    Z.N0star a c = Z.N0star a b + Z.N0star b c := by sorry

Source

https://github.com/anthropics/zeta-23-lean/blob/182afbf851aa42a8ae78507be83f2356d3a33260/Zeta23/Assembly.lean#L912-L918

View graph

Interval additivity of N0∗N_0^*N0∗​

Interval additivity of $N_0^*$