What this theme is
This theme covers the prime-counting function π(x), prime gaps, twin primes, sieve methods, and the prime number theorem with error terms. The size of the error in approximating π(x) by the logarithmic integral is governed precisely by where the zeta zeros lie.
Why it recurs
The distribution of primes is the original motivation for studying ζ(s) at all, so it is never far from view. Sieve methods and gap estimates recur because they translate the analytic content of RH into countable, combinatorial statements about the primes themselves.
Relevance to the Riemann Hypothesis
RH is equivalent to the sharpest possible error term in the prime number theorem: |π(x) − li(x)| = O(√x log x). Every refinement of prime-distribution bounds is, in effect, a probe of how close we are to that optimal estimate.