Shifted Primes and Spectral Detection of Riemann Zeros. Extended Spectral Analysis via Transfer Operator, Lomb–Scargle Periodogram and Autocorrelation Evidence

Source abstract

This paper consolidates, corrects, and extends a research programme on the shifted-prime problem p = q + r - 1 with p, q, r prime, and its connections to the binary Goldbach conjecture and the non-trivial zeros of the Riemann zeta function zeta(s). We collect unconditionally proved results, computationally verified findings with correct methodology, and explicit corrections of errors identified in prior versions. The principal corrections are: (i) the k=3 existence problem is equivalent to the binary Goldbach conjecture and is therefore open, not proved as originally claimed; (ii) scripts 6.py through 8.py contained a critical permutation-test bug that artificially inflated detection counts from 199/200 to the corrected value of 116/200 zeros at p < 0.01 using 709,562 primes in [10^6, 1.2 x 10^7]. New results established in this work include: (iii) the transfer operator ratio lambda1/lambda2 = 182.63 with n = 892,206 primes, nearly three times the value of 62.86 reported previously, confirming systematic spectral strengthening with sample size; (iv) a corrected decay slope of -1.1262 approximately -1 (theoretical prediction under RH), measured consistently across nine independent sample sizes up to n = 1,310,763 primes in [10^6, 2.2 x 10^7], with 129/200 non-trivial Riemann zeros detected at p < 0.01; (v) three new spectral methods, namely autocorrelation in delta-log-p with fundamental period T = 2*pi/gamma1 approximately 0.4445, the Lomb-Scargle periodogram adapted to non-uniform prime spacing, and principal eigenvector analysis of the transfer operator; (vi) three independent analyses, p-adic correlation rho = +0.729, optimal subsequence amplification, and RSA spectral classification, all identify l=3 as the uniquely special prime in the singular factor S_infinity; (vii) the Generalised Amplification Conjecture (GAC) is not yet verifiable in the computational range explored (n <= 1.07 x 10^6 primes), with residual error near 60%. The analytical core, covering absolute convergence of S_infinity, the explicit formula for Psi*(x), equivalence of k=3 to binary Goldbach, and proof of existence for k >= 4 via Helfgott's theorem, remains unconditionally valid and unaffected by the computational corrections. None of these results constitutes a proof of RH. They constitute empirical evidence from an arithmetic direction independent of all classical zero-verification approaches.