We study Möbius inversion along the dilation semigroup $s \mapsto m s$ and package it as an operator calculus. Define the dilation-averaging operator $$(\mathcal{T} G)(s):=\sum_{k \geq 1} \frac{1}{k} G(k s)$$ and its Möbius inverse $$(\mathcal{S} H)(s):=\sum_{m \geq 1} \frac{\mu(m)}{m} H(m s)$$ whenever the series converge locally uniformly. For a nonvanishing holomorphic function $f$ with a coherent branch of $\log f$ along its dilation orbit, the associated prime transform is $$(\mathcal{P} f)(s):=(\mathcal{S} \log f)(s)=\sum_{m \geq 1} \frac{\mu(m)}{m} \log f(m s)$$ and its inverse multiplicative synthesis is $$(\mathcal{E} G)(s):=\exp ((\mathcal{T} G)(s))=\exp \left(\sum_{k \geq 1} \frac{1}{k} G(k s)\right) .$$ Under natural analytic hypotheses, $\mathcal{S} \circ \mathcal{T}=$ Id and $\mathcal{T} \circ \mathcal{S}=$ Id , hence $\mathcal{P}$ and $\mathcal{E}$ are inverses on a corresponding analytic class: $\mathcal{P}(\mathcal{E} G)=G$ and $\mathcal{E}(\mathcal{P} f)=f$. We emphasize a conceptual point: "prime" here means "primitive for the multiplicative factorization carried by $f$," not "the usual integer primes by default." For $\zeta(s)$ one indeed recovers the ordinary primes because $(\mathcal{P} \zeta)(s)$ equals the prime zeta function. For a general $f$, the "primes of $f$ " are the atoms appearing in the spectrum ( $\mathcal{P} f$ ), e.g. as a discrete Laplace/Dirichlet support set. We make this precise first via an atomic-spectrum theorem: if $$G(s)=\sum_j c_j e^{-\lambda_j s},$$ then $\mathcal{E} G$ is a canonical Euler-type product $$\prod_j\left(1-e^{-\lambda_j s}\right)^{-c_j} .$$ We then extend this to measure-valued spectra $$G_\rho(s)=\int_0^{\infty} e^{-\lambda s} d \rho(\lambda)$$ for which $\mathcal{E} G_\rho$ becomes a product integral over a continuous prime spectrum. We also record stability/truncation bounds, singularity propagation under dilation, and Abel damping $\mathcal{P}_\sigma$ as a regularization knob.