A New Identity Connecting the Möbius Function, Euler's Totient Function, and the Golden Ratio with Spectral Properties Related to the Riemann Hypothesis

Here's the Unicode conversion: We present a new identity in analytic number theory connecting the Möbius function μ(k) and Euler's totient function φ(k) through the generating function: ∑_{k=1}^{∞} (μ(k)-φ(k))/k · log(1-x^k) = x²/(1-x) We prove that the equation S(x) = 1 has a unique solution x = φ⁻¹ = (√5-1)/2 in the interval (0,1), where φ is the golden ratio. We extend this to a von Mangoldt-Euler kernel and demonstrate that the associated Dirichlet series exhibits spectral properties directly related to the non-trivial zeros of the Riemann zeta function. Complete asymptotic expansions are derived for two distinct calibrations as x → 1⁻, with the limit π²/6 = ζ(2) rigorously established.