Multiplicative Spectral Geometry: From the Mellin Critical Line to Zeta-Zero Approximation

This work develops a multiplicative geometric framework for the spectral structure of the Riemann zeta function. Starting from harmonic analysis on the multiplicative half-line, the paper shows that the critical line naturally appears as the unique Mellin–Plancherel line associated with unitary multiplicative harmonic analysis. A hyperbolic phase-space geometry generated by the Hamiltonian h(u,τ)=uτ is then used to derive a geometric volume law leading, at the semiclassical level, to Weyl-type asymptotics compatible with the leading growth predicted by the Riemann–von Mangoldt formula. Prime numbers are interpreted through logarithmic prime-power lengths, yielding an orbital distribution whose Laplace transform is identified with the logarithmic derivative of the Riemann zeta function in the half-plane Re(s)>1. The paper also introduces centered truncated Mellin sums and proves, using Euler–Maclaurin summation and Hurwitz’s theorem, that their zeros converge to the non-trivial zeros of the Riemann zeta function. Taken together, these results suggest a coherent framework linking multiplicative harmonic analysis, hyperbolic geometry, orbital arithmetic, and zeta-zero approximation within a common multiplicative spectral geometry.