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An asymptotic formula for the number of totients

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Gives an asymptotic equivalent for the number \(V(x)\) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, \(V(cx)/V(x)\to c\) for every fixed c > 0, answering Erdős and Hall’s scaling question. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

It bears on Erdős problem #416, as listed by Thomas Bloom, erdosproblems.com.

What the papers state

Theorem 2.1

In An asymptotic formula for the number of totients, 25 September 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

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 (2.8) 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 (2.2)–(2.4),

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 2.2

In An asymptotic formula for the number of totients, 25 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

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.

  1. (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).
  2. (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.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.

Reported proof verification
Two Lean statements in the release; one independent group reports that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

An asymptotic formula for the number of totients

25 September 2026, current edition.

Read it herePDF

Let \(V(x)\) count the distinct values of Euler’s totient function up to x. We give an explicit asymptotic equivalent for \(V(x)\). Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that \(V(cx)/V(x)\to c\) as \(x\to\infty\) for every fixed c > 0, answering a question of Erdős and Hall.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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

TotientAsymptotic

The paper states

Theorem 2.1

In An asymptotic formula for the number of totients, 25 September 2026

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 (2.8) 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 (2.2)–(2.4),

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.

The words they share: renewal, ah, nonnegative, formula, real, uniformly, theta, integers, log, asymptotic, totients, arithmetic, finite, cx, positive.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Totient-count asymptotics and regular variation

TotientAsymptotic.lean, the release lists it as a supporting statement.

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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states three more theorems: weighted_totient_asymptotic, weighted_totient_one_two, coefficient_nonnegative.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.

Read the complete Lean statement

TotientCompanionZero

The paper states

Theorem 2.1

In An asymptotic formula for the number of totients, 25 September 2026

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 (2.8) 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 (2.2)–(2.4),

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.

The words they share: arithmetic, finite, fixed, function, positive.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Exact zero case for the companion totient counts

TotientCompanionZero.lean, the release lists it as a supporting statement.

theorem companion_zero_case (k : ℕ) (_hk : 0 < k)    (hno : ¬ ∃ 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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.

Read the complete Lean statement

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

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)

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

Let $V(x)$ count the distinct values of Euler’s totient function up to $x$. 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.

The formalization also gives asymptotics for totients $v\le x$ whose least preimage $\ell(v)=\min\{n\ge1:\varphi(n)=v\}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.

AI-drafted comparisons, with no recorded human review

  • Totient-count asymptotics and regular variation TotientAsymptotic

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (the explicit asymptotic for the number of totients up to x and V(cx)/V(x) → c), together with the asymptotic of the companion theorem for totients whose least preimage lies between kx and (k+1)x.

  • Exact zero case for the companion totient counts TotientCompanionZero

    Unreviewed AI draft: states part of a paper’s theorem. States case (ii) of the companion theorem: when no totient d has least preimage above kd, the count and its coefficient vanish identically. The asymptotic and the positive case are in TotientAsymptotic.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Totient-count asymptotics and regular variation TotientAsymptotic

One report, at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report

Exact zero case for the companion totient counts TotientCompanionZero

One report, at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source

    JSP-000343 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.

  • donteatllamas, 417 (a short write-up deriving a bound from the Erdős 416 result), 10 October 2026. Source

    A consequence of the totient asymptotic gives an explicit upper bound near 2.195 for the limit superior in a related Erdős problem, in a write-up the author says started as AI output and was edited into a short human-readable form; they may also write an exposition of the 416 proof.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare one Lean statement with its paper: TotientAsymptotic, TotientCompanionZeroSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

  • Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source

    A Formal Conjectures maintainer, after source inspection, links the totient asymptotic to part of Erdos problem 416 and leaves another part open.

  • Compiler not recorded (Atlas survey), Atlas survey of Erdős problem pages, recorded on 11 October 2026, 11 October 2026. Source

    The imported Atlas survey reports: Problem 416 page lists two proof claims from before the release (conjectures.io on 27 Sep 2026 and Rafik Zeraoulia on 29 Jul 2026).

  • Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source

    Bloom lists Erdős 416 (counting totient values) and notes the methods resemble Ford’s existing work, with the formalisation linked.

  • mo271, ErdosProblems/416: parts.ii has an explicit asymptotic formula (issue 6907), 7 October 2026. Source

    The release does contain a Lean proof of the totient asymptotic (TotientAsymptotic challenge and UnconditionalMain), but stating the main term needs about 25 definitions (roughly 80 lines), so PR 6898 left part ii open.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 024 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 024 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form