MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
I doubt anything deep lies under a formula containing $\pi(\pi(x))$ and higher compositions (even if one believes that the fractional part of the natural logarithm of something might be interesting). – Greg Martin Jun 9 '13 at 0:42
I would say it is no surprise that one can construct entirely unrelated expressions of that length whose values coincide up to something like 5 decimal digits. -- Just consider the number of possible such expressions and compare it with $10^5$ etc. – Stefan Kohl Jun 9 '13 at 9:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.