Patterns That Eventually Fail

What is Applied Category Theory? Applied Category Theory 2019

Patterns That Eventually Fail

Sometimes patterns can lead you astray. For example, it’s known that

$\displaystyle{ \mathrm{li}(x) = \int_0^x \frac{dt}{\ln t} }$

is a good approximation to $\pi(x),$ the number of primes less than or equal to Numerical evidence suggests that $\mathrm{li}(x)$ is always greater than $\pi(x).$ For example,

$\mathrm{li}(10^{12}) - \pi(10^{12}) = 38,263$

and

$\mathrm{li}(10^{24}) - \pi(10^{24}) = 17,146,907,278$

But in 1914, Littlewood heroically showed that in fact, $\mathrm{li}(x) - \pi(x)$ changes sign infinitely many times!

This raised the question: when does $\pi(x)$ first exceed $\mathrm{li}(x)$ ? In 1933, Littlewood’s student Skewes showed, assuming the Riemann hypothesis, that it must do so for some less than or equal t...