Editing Logarithmic integral function (section)

== Number theoretic significance ==
The logarithmic integral is important in [[number theory]], appearing in estimates of the number of [[prime number]]s less than a given value. For example, the [[prime number theorem]] states that:
: <math>\pi(x)\sim\operatorname{li}(x)</math>
where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>. 

Assuming the [[Riemann hypothesis]], we get the even stronger:<ref>Abramowitz and Stegun, p.&nbsp;230, 5.1.20</ref>
: <math>|\operatorname{li}(x)-\pi(x)| = O(\sqrt{x}\log x)</math>

In fact, the [[Riemann hypothesis]] is equivalent to the statement that:
: <math>|\operatorname{li}(x)-\pi(x)| = O(x^{1/2+a})</math> for any <math>a>0</math>.

For small <math>x</math>, <math>\operatorname{li}(x)>\pi(x)</math> but the difference changes sign an infinite number of times as <math>x</math> increases, and the [[Skewes's number|first time that this happens]] is somewhere between 10<sup>19</sup> and {{val|1.4|e=316}}.