E061: Chebyshev ψ(x) and prime powers¶
Tags: number-theory, quantitative-exploration, visualization, arithmetic-functions, mangoldt, pnt, summatory
See: Valid 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)¶
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=e061uv 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 reportout/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).
n_max: 400000
ψ(n_max) - n_max: +108.161
Figures:
fig_01_psi_vs_x.png
fig_02_psi_minus_x.png
params.json (snapshot)
{
"n_max": 400000
}