# E061: Chebyshev ψ(x) and prime powers

```{figure} ../_static/experiments/e061_hero.png
:width: 80%
:alt: Preview figure for E061
```

```{figure} ../_static/experiments/e061_hero_2.png
:width: 80%
:alt: Preview figure for E061
```

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

## Highlights

- Plot $\psi(x)$ and the error $\psi(x)-x$.
- Make the role of prime powers visible (jumps at p^k).

## Goal

Build intuition for Chebyshev functions as “smoothed” prime counters.

## Background (quick refresher)

- {doc}`../background/von-mangoldt-and-chebyshev`
- {doc}`../background/prime-numbers`

## Research question

Over a moderate range, what does $\psi(x)-x$ look like and where do the main jumps occur?

## Method

- Compute Λ(n) (log p on prime powers) and cumulative ψ(x).
- Plot ψ(x) and ψ(x)-x; optionally annotate largest jumps.

## How to run

- `make run EXP=e061`
- `uv run python -m mathxlab.experiments.e061`

## Outputs

This experiment follows the standard output contract:

- `out/e061/figures/` — generated figures (PNG)
- `out/e061/report.md` — short narrative report
- `out/e061/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/e061.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e061.json
:language: json
```

:::

## References

See {cite:t}`montgomeryvaughan2006multiplicativenumbertheoryi`, {cite:t}`apostol1976introanalyticnumbertheory`.

## Related experiments

- {doc}`e103` (E103: Chebyshev ψ(x): prime powers drive jumps)
- {doc}`e104` (E104: von Mangoldt Λ(n): support and statistics)
- {doc}`e117` (E117: Prime-counting approximations: li(x) and friends)
- {doc}`e054` (Squarefree density via Möbius)
- {doc}`e055` (Mertens function walk)