Lean statement

Totient-count asymptotics and regular variation

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorems selected for verification: OAI.TotientAsymptotic.totient_asymptotic_formula OAI.TotientAsymptotic.weighted_totient_asymptotic OAI.TotientAsymptotic.weighted_totient_one_two OAI.TotientAsymptotic.coefficient_nonnegative

Lean statement file: TotientAsymptotic.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 024 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem totient_asymptotic_formula    :    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))      atTop (Set.Ico (0 : ℝ) 1) ∧    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) := by  sorry⋮theorem weighted_totient_asymptotic    (k : ℕ) (hk : 1≤k) :    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) := by  sorry⋮theorem weighted_totient_one_two    :    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s := by  sorry⋮theorem coefficient_nonnegative (H : ℕ) {s : ℝ}    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.TotientAsymptotic.IsTotient line 13

def IsTotient (v : ℕ) : Prop := ∃ n : ℕ, 0 < n ∧ n.totient = v

definition OAI.TotientAsymptotic.V line 15

def V (x : ℝ) : ℝ := by  classical  exact (((Finset.Icc 1 ⌊x⌋₊).filter IsTotient).card : ℝ)

definition OAI.TotientAsymptotic.ell line 19

def ell (v : ℕ) : ℕ := by  classical  exact if h : IsTotient v then Nat.find h else 0

definition OAI.TotientAsymptotic.a line 23

def a (j : ℕ) : ℝ := (j + 1 : ℝ) * Real.log (j + 1) - j * Real.log j - 1

definition OAI.TotientAsymptotic.RenewalRootSet line 25

def RenewalRootSet : Set ℝ :=  {z | 0 < z ∧ z < 1 ∧ (∑' j : ℕ, a (j+1) * z^(j+1)) = 1}

definition OAI.TotientAsymptotic.rho line 28
```
def rho : ℝ := sInf RenewalRootSet
```
definition OAI.TotientAsymptotic.lam line 30
```
def lam : ℝ := Real.log (1 / rho)
```

definition OAI.TotientAsymptotic.gamma line 32

def gamma : ℝ := (∑' j : ℕ, (j+1 : ℝ) * a (j+1) * rho^(j+1))⁻¹

definition OAI.TotientAsymptotic.g line 34

def g : ℕ → ℝ  | 0 => 1  | n+1 => ∑ d ∈ Finset.range (n+1), a (d+1) * g (n-d)termination_by n => n

definition OAI.TotientAsymptotic.B line 39

def B (x : ℝ) : ℝ := Real.log (Real.log x)

definition OAI.TotientAsymptotic.psi line 40

def psi (b : ℝ) : ℝ := (Real.log b - Real.log (Real.log b)) / lam

definition OAI.TotientAsymptotic.m line 42

def m (x : ℝ) : ℕ := ⌊psi (B x)⌋₊

definition OAI.TotientAsymptotic.theta line 43

def theta (x : ℝ) : ℝ := psi (B x) - m x

definition OAI.TotientAsymptotic.G line 45

def G (x : ℝ) (j : ℕ) : ℝ :=  (B x)^j / ((Nat.factorial j : ℝ) * ∏ i ∈ Finset.Icc 1 j, g i)

definition OAI.TotientAsymptotic.P line 48

def P (H : ℕ) : ℕ := ⌊Real.log (Real.log (H : ℝ))⌋₊

definition OAI.TotientAsymptotic.alpha line 49

def alpha (s : ℝ) : ℝ := lam * Real.exp (lam * s)

definition OAI.TotientAsymptotic.largestPrimeFactor line 51

def largestPrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)

structure OAI.TotientAsymptotic.TailDatum line 53

structure TailDatum (H : ℕ) where  Q : Fin H → ℕ  cofactor : ℕ

definition OAI.TotientAsymptotic.tailPrime line 57

def tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=  if hH : h < H then η.Q ⟨h, hH⟩ else 1

definition OAI.TotientAsymptotic.tailLog line 60

def tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=  Real.log (Real.log (tailPrime η h : ℝ))

definition OAI.TotientAsymptotic.IsWitness line 63

def IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=

Show the remaining 10 lines

  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧  0 < η.cofactor ∧  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧  Real.log (η.cofactor : ℝ) ≤    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)

definition OAI.TotientAsymptotic.w line 75

def w {H : ℕ} (η : TailDatum H) : ℕ :=  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h

definition OAI.TotientAsymptotic.witnesses line 78

def witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=  {η | IsWitness H s η ∧ (w η).totient = d}

definition OAI.TotientAsymptotic.D line 81

def D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l

8 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: renewal, ah, nonnegative, formula, real, uniformly, theta, integers and one more. Found in the paper’s title, which its main theorem states: asymptotic and totients. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: arithmetic, finite, cx and positive.

Theorem 2.1 of An asymptotic formula for the number of totients

Theorem

For sufficiently large positive integers H, there are nonnegative functions A_H(1;\cdot):[0,1)\to\mathbb R, given by the finite arithmetic formula (eq:AH) below, which converge uniformly as H\to\infty to a function A(1;\cdot) satisfying

0<\inf_{0\le s<1}A(1;s) \le\sup_{0\le s<1}A(1;s)<\infty.

With the parameters (eq:renewal)–(eq:Gj),

V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).

Moreover, for every fixed real c>0,

\frac{V(cx)}{V(x)}\longrightarrow c.

Section 2, The explicit main term Read it in the paper TeX source, line 45

Show the TeX

For sufficiently large positive integers $H$, there are nonnegative
functions $A_H(1;\cdot):[0,1)\to\mathbb R$, given by the finite
arithmetic formula \eqref{eq:AH} below,
which converge uniformly as $H\to\infty$ to a function $A(1;\cdot)$
satisfying
\[
 0<\inf_{0\le s<1}A(1;s)
   \le\sup_{0\le s<1}A(1;s)<\infty.
\]
With the parameters \eqref{eq:renewal}--\eqref{eq:Gj},
\[
 V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).
\]
Moreover, for every fixed real $c>0$,
\[
 \frac{V(cx)}{V(x)}\longrightarrow c.
\]

Other possible paper matches, closest wording first

Proposition 6.1 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: max, prime, two, formulas and log.
Proposition 3.3 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: terms, rho, exp, one, witness, uniformly and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: particular.
Theorem 2.2 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: two, ell, uniformly, theta, integer and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
Proposition 5.3 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: factor, totient, ell, tuples and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: particular.
Proposition 6.5 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: coefficient, weights and theta. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Proposition 4.2 of An asymptotic formula for the number of totients. Words of the Lean statement, its names split into words, that also appear in the theorem: count, primes, real, tuples, integer and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive and particular.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 024’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None for the headline (V(cx)/V(x) -> c is stated in terms of V alone). The explicit main term rests on a large custom apparatus that cannot be checked against a standard object from the statement.
Definitions the group asks readers to compare with the paper: Main term built from custom rho := sInf RenewalRootSet, lam, gamma, g (renewal recursion), AH (uses ∑ᶠ finsums over d and over Finset (TailDatum H), whose junk value 0 on infinite support is not excluded by the statement) and A f s := limUnder atTop (fun H => AH H f s) (junk value if no limit; the statement separately asserts TendstoUniformlyOn so it is not vacuous)
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

155 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 119 to its own declarations, 69 to Mathlib’s documentation. 36 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves totient_asymptotic_formula exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 32 other declarations of the file; with the theorem itself they span 105 lines. 11 of them are named in the statement itself.

Names in the namespace OAI.TotientAsymptotic are listed without it.

def IsTotientline 13in the statementlibrary
def Vline 15in the statementlibrary
def ellline 19in the statementlibrary
def aline 23library
def RenewalRootSetline 25library
def rholine 28library
def lamline 30library
def gammaline 32library
def gline 34library
def Bline 39library
def psiline 40library
def mline 42library
def thetaline 43in the statementlibrary
def Gline 45library
def Pline 48library
def alphaline 49library
def largestPrimeFactorline 51library
structure TailDatumline 53library
def tailPrimeline 57library
def tailLogline 60library
def IsWitnessline 63library
def wline 75library
def witnessesline 78library
def Dline 81library
def maxDline 84library
def AHline 88in the statementlibrary
def Aline 102in the statementlibrary
def mainTermline 104in the statementlibrary
def Nline 106in the statementlibrary
def fkline 111in the statementlibrary
def normalizedCountline 113in the statementlibrary
def tupleNormalizationline 116in the statementlibrary
theorem totient_asymptotic_formulaline 118selected theoremlibrary
theorem weighted_totient_asymptoticline 128selected theoremlibrary
theorem weighted_totient_one_twoline 141selected theoremlibrary
theorem coefficient_nonnegativeline 147selected theoremlibrary

import Mathlibnamespace OAI/-! Totient counting and least-preimage asymptotics. -/noncomputable sectionopen scoped BigOperators Topologyopen Filternamespace TotientAsymptoticdef IsTotient (v : ℕ) : Prop := ∃ n : ℕ, 0 < n ∧ n.totient = vdef V (x : ℝ) : ℝ := by  classical  exact (((Finset.Icc 1 ⌊x⌋₊).filter IsTotient).card : ℝ)def ell (v : ℕ) : ℕ := by  classical  exact if h : IsTotient v then Nat.find h else 0def a (j : ℕ) : ℝ := (j + 1 : ℝ) * Real.log (j + 1) - j * Real.log j - 1def RenewalRootSet : Set ℝ :=  {z | 0 < z ∧ z < 1 ∧ (∑' j : ℕ, a (j+1) * z^(j+1)) = 1}def rho : ℝ := sInf RenewalRootSetdef lam : ℝ := Real.log (1 / rho)def gamma : ℝ := (∑' j : ℕ, (j+1 : ℝ) * a (j+1) * rho^(j+1))⁻¹def g : ℕ → ℝ  | 0 => 1  | n+1 => ∑ d ∈ Finset.range (n+1), a (d+1) * g (n-d)termination_by n => ndef B (x : ℝ) : ℝ := Real.log (Real.log x)def psi (b : ℝ) : ℝ := (Real.log b - Real.log (Real.log b)) / lamdef m (x : ℝ) : ℕ := ⌊psi (B x)⌋₊def theta (x : ℝ) : ℝ := psi (B x) - m xdef G (x : ℝ) (j : ℕ) : ℝ :=  (B x)^j / ((Nat.factorial j : ℝ) * ∏ i ∈ Finset.Icc 1 j, g i)def P (H : ℕ) : ℕ := ⌊Real.log (Real.log (H : ℝ))⌋₊def alpha (s : ℝ) : ℝ := lam * Real.exp (lam * s)def largestPrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id)structure TailDatum (H : ℕ) where  Q : Fin H → ℕ  cofactor : ℕdef tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ :=  if hH : h < H then η.Q ⟨h, hH⟩ else 1def tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ :=  Real.log (Real.log (tailPrime η h : ℝ))def IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop :=  (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧  0 < η.cofactor ∧  (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧    (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧    tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧    (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤      (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧  largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧  Real.log (η.cofactor : ℝ) ≤    Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹)def w {H : ℕ} (η : TailDatum H) : ℕ :=  η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η hdef witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) :=  {η | IsWitness H s η ∧ (w η).totient = d}def D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ :=  ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η ldef maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by  classical  exact if ht : T.Nonempty then T.sup' ht (D h) else 0def AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by  classical  exact rho^(H*(H-1)/2) * (gamma / alpha s)^H *    ∑ᶠ d : ℕ,      if IsTotient d then        f ((ell d : ℝ) / d) / d *          ∑ᶠ T : Finset (TailDatum H),            if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then              (-1 : ℝ)^(T.card-1) *               Real.exp (-(gamma / alpha s) *                 ∑' n : ℕ, rho^(H+n) * maxD (H+n) T)            else 0      else 0def A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s)def mainTerm (x : ℝ) : ℝ := x / Real.log x * G x (m x) * A (fun _ => 1) (theta x)def N (k : ℕ) (x : ℝ) : ℝ := by  classical  exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v =>    IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ)def fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r)def normalizedCount (F : ℝ → ℝ) (x : ℝ) : ℝ :=  F x / (x / Real.log x * G x (m x))def tupleNormalization (x : ℝ) : ℝ := x / Real.log x * G x (m x)theorem totient_asymptotic_formula    :    TendstoUniformlyOn (fun H => AH H (fun _ => 1)) (A (fun _ => 1))      atTop (Set.Ico (0 : ℝ) 1) ∧    (∃ cMinus cPlus : ℝ, 0 < cMinus ∧ ∀ s ∈ Set.Ico (0 : ℝ) 1,      cMinus ≤ A (fun _ => 1) s ∧ A (fun _ => 1) s ≤ cPlus) ∧    Tendsto (fun x => V x/mainTerm x) atTop (nhds 1) ∧    ∀ c : ℝ, 0 < c → Tendsto (fun x => V (c*x)/V x) atTop (nhds c) := by  sorrytheorem weighted_totient_asymptotic    (k : ℕ) (hk : 1≤k) :    TendstoUniformlyOn (fun H => AH H (fk k)) (A (fk k)) atTop (Set.Ico (0 : ℝ) 1) ∧    Tendsto (fun x => normalizedCount (N k) x-A (fk k) (theta x)) atTop (nhds 0) ∧    ((∃ d : ℕ, IsTotient d ∧ k*d<ell d) →      ∃ cMinus cPlus : ℝ, 0<cMinus ∧ 0<cPlus ∧        (∀ s∈Set.Ico (0 : ℝ) 1, cMinus≤A (fk k) s ∧ A (fk k) s≤cPlus) ∧        Tendsto (fun x => N k x/(tupleNormalization x*A (fk k) (theta x))) atTop (nhds 1) ∧        ∀ᶠ x : ℝ in atTop, (cMinus/cPlus)*V x≤N k x ∧ N k x≤V x) ∧    ((¬∃ d : ℕ, IsTotient d ∧ k*d<ell d) →      (∀ x : ℝ, 0<x → N k x=0) ∧ (∀ s∈Set.Ico (0 : ℝ) 1, A (fk k) s=0)) := by  sorrytheorem weighted_totient_one_two    :    ∀ k∈({1,2} : Finset ℕ), ∃ c : ℝ, 0<c ∧      ∀ s∈Set.Ico (0 : ℝ) 1, c≤A (fk k) s := by  sorrytheorem coefficient_nonnegative (H : ℕ) {s : ℝ}    (hs : s ∈ Set.Ico (0 : ℝ) 1) : 0 ≤ AH H (fun _ => 1) s := by  sorryend TotientAsymptoticendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

An asymptotic formula for the number of totients

Theorem

0<\inf_{0\le s<1}A(1;s) \le\sup_{0\le s<1}A(1;s)<\infty.

With the parameters (eq:renewal)–(eq:Gj),

V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).

Moreover, for every fixed real c>0,

\frac{V(cx)}{V(x)}\longrightarrow c.

Section 2, The explicit main term Read it in the paper TeX source, line 45

Show the TeX

For sufficiently large positive integers $H$, there are nonnegative
functions $A_H(1;\cdot):[0,1)\to\mathbb R$, given by the finite
arithmetic formula \eqref{eq:AH} below,
which converge uniformly as $H\to\infty$ to a function $A(1;\cdot)$
satisfying
\[
 0<\inf_{0\le s<1}A(1;s)
   \le\sup_{0\le s<1}A(1;s)<\infty.
\]
With the parameters \eqref{eq:renewal}--\eqref{eq:Gj},
\[
 V(x)\sim\frac{x}{\log x}\,G_m A(1;\theta).
\]
Moreover, for every fixed real $c>0$,
\[
 \frac{V(cx)}{V(x)}\longrightarrow c.
\]

Theorem

For each fixed positive integer k, the functions A_H(f_k;s) converge uniformly for 0\le s<1, and

N_k(x)=\frac{x}{\log x}\,G_m\bigl(A(f_k;\theta)+o(1)\bigr).

There are two possibilities.

(i)
If some d\in\mathcal V satisfies \ell(d)>kd, then
\inf_{0\le s<1}A(f_k;s)>0,\qquad N_k(x)\sim\frac{x}{\log x}\,G_m A(f_k;\theta) \asymp_k V(x).
(ii)
If no such d exists, then N_k(x)=0 for every x>0, and A(f_k;s)=0 for every s\in[0,1).

The first alternative holds for k=1,2.

Section 2, The explicit main term Read it in the paper TeX source, line 176

Show the TeX

For each fixed positive integer $k$, the functions $A_H(f_k;s)$ converge uniformly for $0\le s<1$, and
\[
 N_k(x)=\frac{x}{\log x}\,G_m\bigl(A(f_k;\theta)+o(1)\bigr).
\]
There are two possibilities.
\begin{enumerate}[label=\textup{(\roman*)}]
 \item If some $d\in\mathcal V$ satisfies $\ell(d)>kd$, then
 \[
  \inf_{0\le s<1}A(f_k;s)>0,\qquad
  N_k(x)\sim\frac{x}{\log x}\,G_m A(f_k;\theta)
       \asymp_k V(x).
 \]
 \item If no such $d$ exists, then $N_k(x)=0$ for every $x>0$, and $A(f_k;s)=0$ for every $s\in[0,1)$.
\end{enumerate}
The first alternative holds for $k=1,2$.

Configuration and file facts

File: lean/ComparatorChallenges/TotientAsymptotic.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 024 link it.
Solution module: OAI.NumberTheory.TotientAsymptotic.UnconditionalMain
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 32 local vocabulary declarations
Lines of Lean, without comments or blank lines: 105
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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