What this theme is
This theme uses harmonic analysis to study ζ(s): the Mellin transform that links the zeta function to its operator models, Fourier duality between primes and zeros, convolution identities on zero sets, and Weil-style positivity expressed through test functions.
Why it recurs
Fourier and Mellin transforms are the connective tissue between the arithmetic side (primes) and the spectral side (zeros). They recur because nearly every operator construction and explicit formula is, at heart, a transform identity.
Relevance to the Riemann Hypothesis
Weil's explicit formula is a Fourier-analytic identity whose positivity is equivalent to RH. Choosing the right test functions — a harmonic-analysis problem — is one of the most direct attacks on the Hypothesis.