A Complete and Rigorous Proof of the Riemann Hypothesis via Spectral Analysis of Sieve Errors and Ramanujan Summation

We present a complete and rigorous proof of the Riemann Hypothesis. The proof proceeds in three stages. Stage 1 develops a novel sieve framework that detects prime pairs in intervals of the form (p, q2 ), where p and q are consecutive primes. The key innovation is the sieveprimality equivalence theorem, which guarantees that all sieve survivors in this interval are primes, thereby circumventing Selberg’s parity barrier. Using CRT-based residue class constructions, a resonance-breaking double inclusion-exclusion, and symmetric polynomial tail estimates, we prove the Polignac Conjecture and establish the unconditional prime gap bound pn+1 − pn = O(p 1/2+ε n ). The normalized sieve error at this stage satisfies |ap| ≪ p −1+o(1), which is the square-root barrier level. Stage 2 embeds the sieve error into the spectral theory of automorphic forms on GL(2). The indicator function of admissible residue classes is expanded in Dirichlet characters, introducing Gauss sums that connect to Fourier coefficients of Eisenstein series. The crucial technique of complete k-summation converts exponential sums into Ramanujan sums cd(n). Using the divisor representation cd(n) = P e|gcd(d,n) eµ(d/e), we prove the absolute convergence of the resulting double series. After rigorous treatment of the continuous spectrum, discrete spectrum truncation, and contour integral remainders, we obtain the explicit formula 1 ap = X ρ(ρ + 1)p 2β−2 e 2iγ log p + O(p −2+o(1)), ρ breaking the square-root barrier. Stage 3 constructs the spectral detector FP (ω) = 1 π(P) P p≤P papp −iω and proves the fundamental dichotomy: if all zeros lie on the critical line, FP (ω) is uniformly bounded as P → ∞; if there exists an off-critical zero with β0 > 1/2, then |FP (2γ0)| diverges to infinity at rate P 2β0−1/ log P. The final contradiction is obtained by combining the unconditional sieve bound |ap| ≤ Cp−1+ε with the explicit formula. The unconditional bound provides a safe upper limit that any zero contribution must respect, while the explicit formula shows that an offcritical zero would produce a contribution exceeding this limit. The two bounds together yield a contradiction, proving that no off-critical zero can exist. By the functional equation, the same holds for zeros with β < 1/2. Therefore all non-trivial zeros satisfy 34 ℜ(s) = 1/2. Comments and feedback are welcome. Please feel free to contact me via wuhaizhu0512@163.com