What is the value of $\rho$ in the prime counting function $\pi(x)$?

Question

I am trying to write a program which calculates $\pi(x)$, however, the equation requires several instances of the letter $\rho$. I understand that $\rho$ indexes the non-trivial zeroes of the Riemann zeta function, but in the equation, there is no reference to an index of $\rho$, just $\rho$ itself. Please explain what value to give rho, or if it is supposed to somehow be integrated. Thank you.

$$\pi(x)=R(x)-\sum_\rho R\left(x^{\rho}\right)-\frac{1}{\text{ln } x}+\frac{1}{\pi}\text{ arctan }\frac{\pi }{\text{ln } x}$$

Answer 1

You could just do the this - it must be much more efficient:

def primes(N):
  p = range(2,N+1)
  n = 2
  while n*n <= N:
    p = [x for x in p if (x % n) or x <= n]
    n = n + 1
  return p

def pi(N):
  return len(primes(N))

Answer 2

Formula (1) for $\pi(x)$ below can be evaluated as illustrated in formula (2) below where $N\to\infty$ and $K\to\infty$.

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(1) $\quad\pi(x)=R(x)-\sum\limits_\rho R\left(x^{\rho}\right)-\frac{1}{\text{ln } x}+\frac{1}{\pi}\text{ arctan }\frac{\pi }{\text{ln } x}$

(2) $\quad\pi(x)=\left(\sum\limits_{n=1}^N\frac{\mu(n)}{n}\left(\text{li}\left(x^{1/n}\right)-\sum\limits_{k=1}^K\left(\text{Ei}\left(\frac{\log(x)\rho_k}{n}\right)+\text{Ei}\left(\frac{\log(x)\rho_{-k}}{n}\right)\right)\right)\right)\\$ $\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-\frac{1}{\log(x)}+\frac{1}{\pi}\tan^{-1}\left(\frac{\pi}{\log(x)}\right)\\$ $\qquad\qquad\ \ =\left(\sum\limits_{n=1}^N\frac{\mu(n)}{n}\left(\text{li}\left(x^{1/n}\right)-2\,\Re\left(\sum\limits_{k=1}^K\text{Ei}\left(\frac{\log(x)\rho_k}{n}\right)\right)\right)\right)-\frac{1}{\log(x)}+\frac{1}{\pi}\tan^{-1}\left(\frac{\pi}{\log(x)}\right)$

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The following figure illustrates formula (2) above for $\pi(x)$ in orange overlaid on the reference function $\pi(x)$ in blue where formula (2) is evaluated at $N=10$ and $K=200$. I'm a bit skeptical with respect to the convergence of formulas (1) and (2) above for $\pi(x)$ at small values of $x>1$.

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Figure (1): Illustration of Formula (2) for $\pi(x)$ evaluated at $N=10$ and $K=200$ (orange curve)

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