Logarithmic integral function: Difference between revisions

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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{Lili}(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{Lili}(x)-\pi(x) = O(\sqrt{x}\log x)</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 times this happens]] is somewhere between 10¹⁹ and 1.4×10³¹⁶.

== See also ==