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 $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{Li}(x^{1/n})$ deviates from $\pi (x)$ by the asymptotically vanishing sum $\sum_ρ\mbox{Ri}(x^ρ)$ depending on the critical zeros $\rho$ of the Riemann zeta function $\zeta (s)$. We find a fit $π(x)\approx \mbox{Ri}^{(3)}[ψ(x)]$ [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.

6 pages