Fig 78 · Working principle

Primes
leave steps.

Count every prime power, inspect its jump, then choose how much growth your finite ruler allows.

PRIME-POWER PLOTTERREADY
PP / 78
Weighted count ψ(x)dial · ψ/N from0 to1.1
Largest integer residual R
ψ(x)−xSelected R

Count window: ψ/N and x/N, fixed y0–1.1. Residual window: integer R samples and C, fixed y0–1. in both windows.

What these finite traces mean

ψ(x)=Σ log(p) over all prime powers p^k≤x, with natural logs. Powers such as64=2^6 add log2. At a power the count includes the jump; other integers add zero. This is weighted counting, not π(x).

R(x)=|ψ(x)−x|/x^β. The largest R across integers16…N is the least C covering this finite set with Cx^β. Right-window connections join discrete samples; noninteger extrema and x outside16…N are not checked.

Changing N changes the labelled x interval and count normalization, not either y ruler. Dial scales are ψ/N0–1.1 and maximum R0–1. Cursor changes clamp to N; manual sweeps stop at the range end.

β.875 is a growth ruler, not a verified zero-free region. Constants, logarithmic factors and asymptotic quantifiers matter. No zeta zeros, Dirichlet uniformity, RH/quasi-RH, paper or Lean proof are checked here.

NIST DLMF: weighted prime counts and asymptotic context ↗
NIST DLMF: arithmetic functions ↗