Efficient prime counting and the Chebyshev primes

Abstract

The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta (x)={\sum }_{p\le x}\log p$ and $\psi (x)={\sum }_{n=1}^x\Lambda (n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $\mu (n)$ and $\Lambda (n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, ${\epsilon }_{\theta }$ and ${\epsilon }_{\psi }$ may potentially occur only at $x+1\in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{\theta (p)}$ and $j_{\psi (p)}$. In particular, $j_p<1$, and $j_{\theta (p)}>1$ for $p<10^{11}$. Besides, $j_{\psi (p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109,113,139,181,197,199,241,271,281,283,293,313,317,443,449,461,463,\dots \}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{\psi (p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $\psi (x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi (p_n^l)-p_n^l$ (or of the function $\theta (p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{li}[ψ(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.

15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are new