Improving riemann prime counting

Abstract

Prime number theorem asserts that (at large $x$) the prime counting function $\pi(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{Li}(x^{1/n})$ deviates from $\pi(x)$ by the asymptotically vanishing sum $\sum_{\rho}\mbox{Ri}(x^\rho)$ depending on the critical zeros $\rho$ of the Riemann zeta function $\zeta(s)$. We find a fit $\pi(x)\approx \mbox{Ri}^{(3)}[\psi(x)]$ [with three to four new exact digits compared to $\mbox{li}(x)$] by making use of the Von Mangoldt explicit formula for the Chebyshev function $\psi(x)$. Another equivalent fit makes use of the Gram formula with the variable $\psi(x)$. Doing so, we evaluate $\pi(x)$ in the range $x=10^i$, $i=[1\cdots 50]$ with the help of the first $2\times 10^6$ Riemann zeros $\rho$. A few remarks related to Riemann hypothesis (RH) are given in this context.