E063: Dirichlet convolution playground

Preview figure for E063

Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, dirichlet-convolution, model-checking See: Valid Tags.

Highlights

  • Compute Dirichlet convolutions numerically on a finite range.

  • Check exact identities and report any mismatches (should be zero).

Goal

Turn algebraic identities of arithmetic functions into concrete computed checks.

Background (quick refresher)

Research question

Can we numerically verify standard Dirichlet-convolution identities up to a bound without mistakes?

Method

  • Implement divisor-sum convolution on [1..N].

  • Compute μ1 and μid and compare to ε and φ respectively.

How to run

  • make run EXP=e063

  • uv run python -m mathxlab.experiments.e063

Outputs

This experiment follows the standard output contract:

  • out/e063/figures/ — generated figures (PNG)

  • out/e063/report.md — short narrative report

  • out/e063/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: 60000

Checks:

  • μ * 1 = ε : OK

  • μ * id = φ : OK

Figure:

  • fig_01_abs_error.png

params.json (snapshot)
{
  "n_max": 60000
}

References

See Apostol [1976].