This paper studies a compact integral operator on $L^2(R+,dx/x) $ whose kernel encodes arithmetic structure through prime logarithmic frequencies. We introduce the Euler Arithmetic Operator, defined via logarithmic convolution with a kernel constructed from prime power phases modulated by an even window function. We prove two main results: (1) the spectrum of this operator is invariant under multiplicative phase twisting $ψ(x)↦x^{iθ}ψ(x)$ showing that its eigenvalues are intrinsic and independent of logarithmic frame; and (2) the spectrum is given explicitly as the range of the Fourier transform of the kernel, expressed as a superposition of shifted spectral profiles located at the frequencies $kln⁡p$. The result identifies the spectrum of the Euler Arithmetic Operator as a canonical multiplicative spectral invariant.