Consider that if a number is \[N = \prod_i p_i^{q_i}\] with primes \(p\) and some integer powers \(q\). Then \[log(N)=\sum_j q_jlog(p_j)\]
Now if all the \(p_j\) are orthogonal in some sense, and the infinite number of primes form a Hilbert space, we have a loose anology as the log of a number being a wavefunction as \[\Psi(x,t) = \sum_n a_n\psi_n(x,t)\]
In the reverse we have \[e^{\Psi(x,t)} = \prod_n e^{a_n\psi_n(x,t)}\]
If \(\Psi\) and \(\psi\) are complex valued, we then have a seperation of parts \[\Psi(x,t) = R(x,t) +i\Upsilon(x,t) \\ \psi(x,t) = r(x,t) +iu(x,t) \\ e^{R(x,t)}(\mathrm{cos}(\Upsilon(x,t))+i\mathrm{sin}(\Upsilon(x,t)) = \prod_n e^{a_nr_n(x,t)} (\mathrm{cos}(a_nu_n(x,t))+i\mathrm{sin}(a_nu_n(x,t))\]