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 $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$, $ε_θ$ and $ε_ψ$ 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 $j_p$, $j_{θ(p)}$ and $j_{ψ(p)}$. In particular, $j_p<1$, and $j_{θ(p)}>1$ for $p<10^{11}$. Besides, $j_{ψ(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{ψ(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $ψ(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $ψ(p_n^l)-p_n^l$ (or of the function $θ(p_n^l)-p_n^l$ ). 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.