Result 024, Number theory

An asymptotic formula for the number of totients

Gives an asymptotic equivalent for the number V(x)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)→cV(cx)/V(x)\to c for every fixed c > 0, answering Erdős and Hall’s scaling question.

Lean formalization Proof

The bigger picture

Why it matters

Euler's totient function counts how many positive integers up to a given integer share no common divisor with it except one. The manuscript describes how many different answers this counting rule produces below a growing cutoff.

What changes?

Let V(x) count distinct totient values at most x, counting each value once even if several integers produce it. The unreviewed manuscript reports an explicit asymptotic equivalent: a formula whose ratio to V(x) tends to one as x grows. It includes a positive, bounded, phase-dependent factor. The coefficient is determined by a uniform limit of functions built from finite arithmetic data, connecting the limiting formula to convergent arithmetic approximations.

What does that help mathematicians do?

The reported scaling law says that, for every fixed positive c, V(cx) divided by V(x) tends to c as x grows. This answers Erdős and Hall's question and controls counts across proportional intervals. For example, the number of distinct totient values between x and twice x is asymptotically equal to the number up to x, despite the formula's phase-dependent factor.

Are there practical applications?

The immediate value is foundational: the result gives number theorists a precise large-scale description of the set of possible totient outputs. It supports comparisons of how many such values occur below different cutoffs. It does not, by itself, identify which individual integers are totients or how many inputs produce each value.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

An asymptotic formula for the number of totients

September 25, 2026 39 pages

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

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026,
  author = {{OpenAI}},
  title = {{An asymptotic formula for the number of totients}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf}{OAI:An-asymptotic-formula-for-the-number-of-totients-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/024.md.

An asymptotic formula for the number of totients

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

Let V(x)V(x) count the distinct values of Euler's totient function up to xx. 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)→cV(cx)/V(x)\to c for every fixed c>0c>0, answering the Erdős–Hall regular-variation question.

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

Comparator links

Result Comparator statement
Totient-count asymptotics and regular variation TotientAsymptotic.lean
Exact zero case for the companion totient counts TotientCompanionZero.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 12 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.