Summary: The action of \(\mathbb{N}\) on \(\ell^2(\mathbb{N})\) is studied in association with the multiplicative structure of \(\mathbb{N}\). Then the maximal ideal space of the Banach algebra generated by \(\mathbb{N}\) is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The \(C^*\)-algebra generated by \(\mathbb{N}\) does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by \(\mathbb{N}\) is \(B(\ell^2(\mathbb{N}))\), the set of all bounded operators on \(\ell^2(\mathbb{N})\). Moreover, the differential operator on \(\ell^2(\mathbb{N}), \frac1{n(n+1)})\) defined by \(\nabla f = \mu^* f\) is considered, where \(\mu\) is the Möbius function. It is shown that the spectrum \(\sigma(\nabla)\) contains the closure of \(\{\zeta(s)^{-1}: \operatorname{Re}(s) > 1\}\). Interesting problems concerning \(\nabla\) are discussed.