Image: Prime number theorem ratio convergence
Description: A plot showing how two estimates described by the prime number theorem, xlnx{\displaystyle {\frac {x}{\ln x}}} and ∫2x1lntdt=Li(x)=li(x)−li(2){\displaystyle \int _{2}^{x}{\frac {1}{\ln t}}\mathrm {d} t=Li(x)=li(x)-li(2)} converge asymptotically towards π(x){\displaystyle \pi (x)}, the number of primes less than x. The x axis is x{\displaystyle x} and is logarithmic (labelled in evenly spaced powers of 10), going up to 1024, the largest x{\displaystyle x} for which π(x){\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.
Title: Prime number theorem ratio convergence
Credit: Own work
Author: Dcoetzee
Usage Terms: Creative Commons Zero, Public Domain Dedication
License: CC0
License Link: http://creativecommons.org/publicdomain/zero/1.0/deed.en
Attribution Required?: No
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