# Totient counting asymptotic reference

Family 024: An asymptotic formula for the number of totients. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic database researchers can compare distinct totient counts to a phase-dependent asymptotic equivalent.

## Applicability and commercial boundary

The bounded factor is not a constant, and convergent arithmetic approximations may be expensive. No efficient inverse-totient enumerator follows.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Inspect computable factor approximation cost and finite error.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[An asymptotic formula for the number of totients](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf).