We study Mellin transforms $\widehat{N}\left(s\right)={\int }_{1-}^{\infty }{x}^{-s}dN\left(x\right)$ for which $N\left(x\right)-x$ is periodic with period $1$ in order to investigate ‘flows’ of such functions to Riemann’s $\zeta \left(s\right)$ and the possibility of proving the Riemann Hypothesis with such an approach. We show that, excepting the trivial case where $N\left(x\right)=x$, the supremum of the real parts of the zeros of any such function is at least $\frac{1}{2}$.We investigate a particular flow of such functions ${\left\{\widehat{N_{\lambda }}\right\}}_{\lambda \ge 1}$ which converges locally uniformly to $\zeta \left(s\right)$ as $\lambda \to 1$, and show that they exhibit features similar to $\zeta \left(s\right)$. For example, $\widehat{N_{\lambda }}\left(s\right)$...