What this theme is
The Riemann–von Mangoldt explicit formula expresses prime-counting functions as a sum over the zeta zeros. Zero-free regions — strips near Re(s) = 1 proven to contain no zeros — are the partial results from which all unconditional prime estimates follow. Papers here sharpen these formulae and widen the cleared region.
Why it recurs
The explicit formula is the bridge that makes every other theme possible: it is literally the identity that turns facts about zeros into facts about primes. Zero-free regions recur as the concrete, provable substitute for the full strength of RH.
Relevance to the Riemann Hypothesis
RH says the entire critical strip except the line Re(s) = 1/2 is zero-free. Each widening of the known zero-free region is a step toward that goal, and the explicit formula quantifies exactly what each step buys.