# E057: Erdős–Kac in practice

```{figure} ../_static/experiments/e057_hero.png
:width: 80%
:alt: Preview figure for E057
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `arithmetic-functions`, `omega`, `heuristics`
See: {doc}`../tags`.

## Highlights

- Compute $\Omega(n)$ for $n\le N$ and plot its histogram.
- Optionally normalize to compare with a Gaussian curve (Erdős–Kac heuristic).

## Goal

Visualize the probabilistic number-theory flavor of prime-factor counts.

## Background (quick refresher)

- {doc}`../background/omega-functions`
- {doc}`../background/average-orders-and-erdos-kac`

## Research question

How close does the distribution of $\Omega(n)$ (after normalization) look to a normal curve for moderate $N$?

## Method

- Compute $\Omega(n)$ using an SPF sieve.
- Plot histogram; optionally overlay a normal density with matching mean/variance approximations.

## How to run

- `make run EXP=e057`
- `uv run python -m mathxlab.experiments.e057`

## Outputs

This experiment follows the standard output contract:

- `out/e057/figures/` — generated figures (PNG)
- `out/e057/report.md` — short narrative report
- `out/e057/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).

```{include} ../reports/e057.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

::: {dropdown} params.json (snapshot)
:open:

```{literalinclude} ../params/e057.json
:language: json
```

:::

## References

See {cite:t}`erdoskac1940gaussianlawerrorsadditivefunctions`, {cite:t}`tenenbaum2015introanalyticprobabilisticnumbertheory`.

## Related experiments

- {doc}`e094` (E094: ω(n) vs. Ω(n): Erdős–Kac normalization)
- {doc}`e095` (E095: Squarefree filter: ω(n)=Ω(n) when μ(n)≠0)
- {doc}`e118` (E118: Chebyshev bias: lead-time statistics)
- {doc}`e120` (E120: Liouville λ(n): partial sums and parity)
- {doc}`e052` (Totient ratio landscape)