Consider the series defining the exponential function $\displaystyle{\sum_{k=0}^{\infty}\dfrac{x^{k}}{k!}}$ and replace $x^k$ by $f^{(k)}$ where $f^{(0)}=Id$ and $f^{(k+1)}=f\circ f^{(k)}$. Applying this to $\pi$, the counting function of primes, one gets the function $\displaystyle{E_{\pi}:x\mapsto\sum_{k=0}^{\infty}\dfrac{\pi^{(k)}(x)}{k!}}$.

When one tries to evaluate $\dfrac{E_{\pi}(100)-100}{100}$, one gets $0.30258333333...$, whereas $\{\log 10\}=0.30258509...$.

Is this just a coincidence or does something deeper lie under this numerical agreement?

Thanks in advance.