# E071: PNT(AP) numerics: pi(x;q,a) - Li(x)/phi(q).

```{figure} ../_static/experiments/e071_hero.png
:width: 80%
:alt: Preview figure for E071
```

**Tags:** `number-theory`, `quantitative-exploration`, `visualization`, `aps`

## Highlights
- Counts primes in residue classes a mod q for several a.
- Compares empirical counts to the first-order prediction li(x)/φ(q).
- Tracks a simple error proxy across x to visualize deviation patterns.

## What this experiment does

The prime number theorem in arithmetic progressions suggests:

The implementation focuses on a compact, reproducible numerical workflow: deterministic parameter defaults, structured output folders, and one or more figures saved for the gallery.

## Outputs

This experiment writes into `out/e071/`:

- `figures/fig_01_error_terms.png`

## How to run

```bash
make run EXP=e071
```

## Notes

- The gallery preview figure shipped with the documentation uses conservative cutoffs so builds stay fast. If you run the experiment locally, increase the cutoffs to see the asymptotic regime more clearly.
- Prime-race plots depend on the chosen sampling of `x` (linear vs. log grid). The qualitative “who leads” picture can change when you zoom in.

## Published run snapshot

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

```{include} ../reports/e071.md
:start-after: "<!-- REPORT:BEGIN -->"
:end-before: "<!-- REPORT:END -->"
```

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

```{literalinclude} ../params/e071.json
:language: json
```

:::

## References

{cite:t}`apostol1976introanalyticnumbertheory, davenport2000multiplicativenumbertheory`

## Related experiments

- {doc}`e070` (Primes in residue classes: pi(x; q, a).)
- {doc}`e072` (Prime race mod 4: pi(x;4,3) vs. pi(x;4,1).)
- {doc}`e073` (Prime race mod 3: pi(x;3,2) vs. pi(x;3,1).)
- {doc}`e074` (Prime race mod 8: leaderboard among 1,3,5,7.)
- {doc}`e075` (Prime race statistic: distribution on a log-grid.)