I'm trying to spark some interest in discussion in prime number theory and the Riemann Zeta Function/Hypothesis. The prime counting function, $\displaystyle \pi(x)$, outputs the number of primes less than or equal to x. Gauss suggested at 15 that a good approximation for this prime counting function was $\displaystyle Li(x)$, where $\displaystyle Li(x)=\int_{0}^{\infty}\frac{dx}{\log(x)}$ (I realize that log(1)=0). This is a good approximation and although it seems to always bound the prime counting function, they do eventually cross.

Anyway, here is an explicit formula for a prime counting function.

$\displaystyle J(x)=Li(x)-\sum_{\rho}\frac{x^{\rho}}{\rho}-\ln(2)+\int_{x}^{\infty}\frac{dt}{t(t^2-1}\ln(t)}$

Here $\displaystyle \rho$ represents the non-trivial zeros of the Riemann Zeta Function in the critical strip. Since there are infinitely many zeros in the critical strip this formula is not feasible to use for actual caculations, but my question is that if we could use this formula would it output integers? I mean would it sayexactlyhow many primes are less than or equal to x?

I'd appreciate any other thoughts to add to my own.