Prime number theorem ratio convergence

Författare/Upphovsman:

Dcoetzee

Kreditera:

Eget arbete

Kort länk:

https://sewiki.info/b/8c3d5

Källa:

Wikimedia Commons

Upplösning:

250 x 160 Pixel (88576 Bytes)

Beskrivning:

A plot showing how two estimates described by the prime number theorem, ${\displaystyle {\frac {x}{\ln x}}}$ and ${\displaystyle \int _{2}^{x}{\frac {1}{\ln t}}\mathrm {d} t=Li(x)=li(x)-li(2)}$ converge asymptotically towards ${\displaystyle \pi (x)}$, the number of primes less than x. The x axis is ${\displaystyle x}$ and is logarithmic (labelled in evenly spaced powers of 10), going up to 10²⁴, the largest ${\displaystyle x}$ for which ${\displaystyle \pi (x)}$ is currently known. The former estimate converges extremely slowly, while the latter has visually converged on this plot by 10⁸. Source used to generate this chart is shown below.

Licens:

CC0

Licensvillkor:

Creative Commons Zero, Public Domain Dedication