The zeta process

A random natural number Z_s with the zeta distribution, built prime by prime. For each prime p a stack of independent clocks E_p,1, E_p,2, … (rate log p); the exponent of p in Z_s is the length of the leading run that beats the threshold s. Slide s down the vertical axis (log-scaled in 1/(s−1)) and Z_s traces a divisibility chain — jumping by whole prime powers, so it can leap over numbers. Press Run to accumulate instantiations and watch the histogram of Z_s fill toward 1/(ζ(s) n^s). From §10.2 of Alexeev–Barreto–Li–Lichtman–Price–Shah–Tang–Tao.

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Z_s = — s = —

Drag anywhere on the top plot to set s (this resets the histogram; the ζ(s) curve tracks s live). Green markers below the line are the clocks still beating it; each chain node is a bubble at the exact s where Z_s jumps to it. Faded bubbles are recent instantiations.

What you are looking at

Each clock E_p,k is exponential with rate log p, so P(E_p,k ≥ s) = p^−s. The exponent of p in Z_s is e_p,s = the length of the initial run E_p,1, …, E_p,e ≥ s, a geometric variable, and Z_s = ∏_p p^e_p,s has the zeta distribution P(Z_s = n) = 1/(ζ(s) n^s) — the bars below converge to it.

For one fixed set of clocks, lowering s only lengthens the runs, so Z_s grows through divisibility: Z_s | Z_t for t < s. The jumps happen at running-minimum thresholds and multiply by a whole prime power p^j — exactly the von Mangoldt downward chain — so the chain can jump over numbers (e.g. straight from 1 to 4). Runs entirely in your browser; no network.