Pretentious number theory refresher¶
“Pretentious” number theory replaces some zero-driven arguments by measuring how closely a multiplicative function pretends to be a simple model (typically \(n^{it}\) or a Dirichlet character). It is especially effective for comparative experiments: which model explains the data best?
Core definitions¶
Let \(f,g\) be multiplicative functions bounded by 1 in magnitude on primes. The pretentious distance up to \(x\) is
Interpretation:
\(\mathbb{D}(f,g;x)\) small means \(f(p)\) and \(g(p)\) “agree” on most primes (in a \(1/p\)-weighted sense).
\(\mathbb{D}(f,g;x)\) large means \(f\) cannot be explained well by \(g\) on primes.
A common model family is \(g(n)=\chi(n)n^{it}\) with a Dirichlet character \(\chi\) and real \(t\).
What experiments usually visualize or measure¶
Fit \(t\) to minimize \(\mathbb{D}(f,n^{it};x)\) and plot the best-fit \(t(x)\).
Compare distances \(\mathbb{D}(f,\chi;x)\) across characters \(\chi\) to see which congruence structure matches \(f\).
Use the distance to explain why partial sums of \(f(n)\) may be large or small.
Practical numerical caveats¶
Always define the prime cutoff (use the same \(x\) for fair comparisons).
Distance is dominated by small primes; if you want to study “large-prime behavior,” consider plotting contributions by prime ranges.
When fitting \(t\), the objective can have shallow minima; report robustness (e.g., multiple initial guesses).
References¶
See References.
Experiments in this repository¶
E120 — Pretentious distance atlas for core multiplicative functions (μ, λ, φ/n, …).