Improving riemann prime counting

TL;DR

This paper improves the approximation of the prime counting function using Riemann zeros and explicit formulas, achieving higher accuracy in large ranges and discussing implications for the Riemann hypothesis.

Contribution

It introduces a novel fit for al(x) using the Riemann prime counting function and the Chebyshev al function, enhancing accuracy with new exact digits.

Findings

Achieved three to four new exact digits in prime counting approximation.

Evaluated al(x) up to 10^50 using 2 million Riemann zeros.

Discussed implications of results for the Riemann hypothesis.

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.