What this theme is
This theme models the primes or the zeros as random processes: Brownian motion and stochastic calculus, statistical-mechanics partition functions, entropy and thermodynamic formalism, and probabilistic heuristics like Cramér's model. The de Bruijn–Newman constant, governed by a heat flow on the zeros, also appears here.
Why it recurs
The primes behave randomly even though they are deterministic, so probabilistic and thermodynamic language repeatedly captures their statistics where exact formulae are unavailable. Stochastic stability arguments mirror the eigenvalue-repulsion picture from the random-matrix theme.
Relevance to the Riemann Hypothesis
The de Bruijn–Newman constant Λ satisfies Λ ≤ 0 if and only if RH holds, and it is now known that Λ ≥ 0; so RH is exactly the statement Λ = 0. Probabilistic models aim to show the zeros are statistically pinned to the critical line.