Proof of the Riemann Hypothesis,Version 3

We prove the Riemann Hypothesis by establishing that all non-trivial zeros of the Riemann zeta function ζ(s) satisfy ℜ(s) = 1/2. The proof constructs a local sieve framework for prime pairs in the interval (p, q2 ), where q is the prime following p. The key innovation is the discovery that within this specific interval, the sieve condition is exactly equivalent to primality—a consequence of the fact that any composite number in (p, q2 ) must have a prime factor not exceeding p. This exact equivalence allows us to connect the sieve count directly to the Goldston-Montgomery explicit formula for prime pairs without any approximation. Using the product structure of the admissible residue classes modulo small primes and a detailed analysis of the distribution of gcd(k, M0), we obtain an unconditional error bound of p −1+o(1) for the normalized sieve error. This bound is sufficiently sharp to detect any hypothetical zero with ℜ(s) > 1/2 via its deterministic contribution to the sieve error. Comparing this contribution with the unconditional error bound yields an immediate contradiction unless all zeros satisfy ℜ(s) ≤ 1/2. The functional equation of ζ(s) then forces ℜ(s) = 1/2 for all non-trivial zeros. The proof is entirely unconditional, relying only on rigorously established results in analytic number theory including Mertens’ theorems, Parseval’s identity, the explicit formula for Ramanujan sums, and the Guinand-Weil explicit formula. Comments and feedback are welcome. Please feel free to contact me via [email protected]