Exploratory visualizations for arithmetic functions¶
Several experiments in this project are intentionally “EDA-like”: they produce heatmaps, atlases, and correlation matrices for many functions at once. This page collects best practices so those figures remain interpretable and comparable.
Core definitions¶
Given a function table \(F(n)\) for \(n=1,\dots,N\) and a list of arithmetic functions \(f_1,\dots,f_m\), two common derived views are:
Heatmap atlas: visualize values \(f_j(n)\) as an image (rows = functions, columns = \(n\) or blocks of \(n\)).
Correlation matrix: compute an empirical correlation between columns \(f_i(n)\) and \(f_j(n)\) (Pearson or rank-based).
What experiments usually visualize or measure¶
“Texture”: squarefree bands, prime powers, smooth numbers, record-holders, etc.
Clustering: which functions behave similarly on \(1..N\) (after normalization).
Stability: how patterns change when \(N\) grows or when you sample on a log grid.
Practical numerical caveats¶
Normalize first. Raw scales differ wildly (e.g. \(\sigma(n)\) vs \(\mu(n)\)). Typical choices: z-score, log1p, or scaling by \(n^\alpha\).
Beware heavy tails. Extremal values can dominate Pearson correlation; consider Spearman correlation or winsorization for robustness.
Sampling matters. Linear \(n\) emphasizes small structure; log-spaced \(n\) emphasizes asymptotic behavior. Record the sampling rule in the figure caption.
References¶
See References.
Experiments in this repository¶
E122 — Heatmap atlas of μ, ω, Ω and related “squarefree texture” features.
E123 — Correlation matrix of arithmetic functions (with normalization variants).