Logarithmic Phases and Arithmetic Weights in a Discrete Spectral Exploration of the Riemann Zeta Zeros

This note presents an exploratory discrete spectral model for the nontrivial zeros of the Riemann zeta function. The starting idea is to reinterpret the Dirichlet-type oscillatory structure behind ∑n−(σ+it)\sum n^{-(\sigma+it)}∑n−(σ+it) as an interference amplitude generated by logarithmic phases. For each truncation size NNN, a finite-dimensional Hamiltonian Ht(N)H_t^{(N)}Ht(N) is constructed, and the resonance function RN(t)=min⁡j∣λj(Ht(N))∣R_N(t)=\min_j |\lambda_j(H_t^{(N)})|RN(t)=minj∣λj(Ht(N))∣ is used to detect quasi-resonant states. Deep minima of RN(t)R_N(t)RN(t) are then compared numerically with the imaginary parts of the nontrivial zeros of ζ(1/2+it)\zeta(1/2+it)ζ(1/2+it). The initial model, based on diagonal terms log⁡n\log nlogn and off-diagonal couplings of the form cos⁡(tlog⁡(n/m))/nm\cos(t\log(n/m))/\sqrt{nm}cos(tlog(n/m))/nm, did not produce sufficiently selective minima near the zeta zeros. After introducing a reduced diagonal scaling parameter and testing a generalized coupling exponent, the most promising version was obtained by incorporating an arithmetic weight based on gcd⁡(n,m)\gcd(n,m)gcd(n,m). The best-performing Hamiltonian used diagonal entries 0.05log⁡n0.05\log n0.05logn and off-diagonal terms gcd⁡(n,m)/(nm)⋅cos⁡(tlog⁡(n/m))\gcd(n,m)/(nm)\cdot \cos(t\log(n/m))gcd(n,m)/(nm)⋅cos(tlog(n/m)). In this form, the model produced stable numerical associations for the first four nontrivial zeros, with a total error of 0.4924, and remained numerically stable for truncation sizes N=80,120,160,200N=80,120,160,200N=80,120,160,200. When extended to the first eight zeros, the total error increased to 2.2329, indicating that the model captures a genuine arithmetic signal but does not provide a faithful global description. The main outcome is therefore not a proof of the Riemann Hypothesis, but an exploratory framework suggesting that logarithmic phases, spectral resonance, and authentic arithmetic weights may deserve further investigation in discrete operator models related to zeta phenomena