# Regarding the prime counting function

I'm certain that you are aware of the fact that a number n is a prime only if

$$p(n) =\frac{\Gamma(n){^n}}{n} \notin \Bbb N$$

Now consider the function

$$\phi(x) = \begin{cases}1, & \text{if x \notin \Bbb N}\\ 0, &\text{if x \in \Bbb N} \end{cases} = \lceil x - \lfloor x\rfloor \rceil$$

Using the above two functions, we have,

$$Q(x) = \begin{cases}1, & \text{if x is a prime}\\ 0, &\text{otherwise } \end{cases} = \phi(p(x))$$

then the prime counting function $\pi(n)$ will be,

$$\pi(n) = \sum_{i= 2}^n \phi(p(i)) = \left\lceil \frac{\Gamma(i){^i}}{i} - \left\lfloor \frac{\Gamma(i){^i}}{i}\right\rfloor \right\rceil$$

Am I correct? Is there any way to proceed furthur?

• There are many other (much simpler IMO) ways for describing $\pi(n)$, as well as $p_n$ (the prime number sequence) BTW. The real challenge is to establish a prime-formula which is not "computationally worthless". – barak manos Sep 5 '16 at 8:18
• Nice consideration - but to proceed here means wasting the time. – user90369 Sep 5 '16 at 8:34