E060: Jordan totients

Preview figure for E060

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, totient, multiplicative See: Valid Tags.

Highlights

  • Compute \(J_k(n)\) for k=1,2,3 and plot \(J_k(n)/n^k\).

  • Compare how the small-prime structure changes as k increases.

Goal

Show a clean generalization of φ(n) and how normalization reveals multiplicative structure.

Background (quick refresher)

Research question

How do the normalized Jordan totients \(J_k(n)/n^k\) behave for k=1..3?

Method

  • Compute \(J_k(n)=n^k\prod_{p\mid n}(1-p^{-k})\) via prime factors.

  • Plot normalized values for k=1..3.

How to run

  • make run EXP=e060

  • uv run python -m mathxlab.experiments.e060

Outputs

This experiment follows the standard output contract:

  • out/e060/figures/ — generated figures (PNG)

  • out/e060/report.md — short narrative report

  • out/e060/manifest.json — snapshot metadata for the gallery

Published run snapshot

If this experiment is included in the docs gallery, include the published snapshot (report + params).

  • n_max: 80000

Figure:

  • fig_01_jordan_normalized.png

params.json (snapshot)
{
  "n_max": 80000
}

References

See Apostol [1976].