New fast methods to compute the number of primes less than a given value

• Datum: 2022
• 1. Verfasser: Teruel, G. R. P.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Institute of Mathematics, NAS of Ukraine 2022
• Online Zugang: https://umj.imath.kiev.ua/index.php/umj/article/view/6193
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.688 The paper describes new fast algorithms for evaluating $\pi(x)$ inspired by the harmonic and geometric mean integrals that can be used on any pocket calculator.  In particular, the formula $h(x)$ based on the harmonic mean is within $\approx 15$ of the actual value for $3\leq x\leq 10000.$ The approximation verifies the inequality, $h(x)\leq {\rm Li}(x)$ and, therefore, is better than ${\rm Li}(x)$ for small $x.$  We show that $h(x)$ and their extensions are more accurate than other famous approximations, such as Locker–Ernst's or Legendre's also for large $x.$  In addition, we derive another function $g(x)$ based on the geometric mean integral that employs $h(x)$ as an input, and allows one to significantly improve the quality of this method.  We show that $g(x)$ is within $\approx 25$ of the actual value for $x\leq 50000$ (to compare ${\rm Li}(x)$ lies within $\approx 40$ for the same range) and asymptotically $g(x)\sim \dfrac{x}{\ln x}\exp\left(\dfrac{1}{\ln x-1}\right).$