Efficient prime counting and the Chebyshev primesReport as inadecuate

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1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies 2 Télécom ParisTech

Abstract : The function $\epsilonx=\mbox{li}x-\pix$ is known to be positive up to the very large Skewes- number. Besides, according to Robin-s work, the functions $\epsilon {\theta}x=\mbox{li}\thetax-\pix$ and $\epsilon {\psi}x=\mbox{li}\psix-\pix$ are positive if and only if Riemann hypothesis RH holds the first and the second Chebyshev function are $\thetax=\sum {p \le x} \log p$ and $\psix=\sum {n=1}^x \Lambdan$, respectively, $\mbox{li}x$ is the logarithmic integral, $\mun$ and $\Lambdan$ 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 $j p$, $j {\thetap}$ and $j {\psip}$. In particular, $j p1$ for $p

Author: Michel Planat - Patrick Solé -

Source: https://hal.archives-ouvertes.fr/


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