Logarithmic integral function: Difference between revisions

Revision as of 19:55, 11 December 2022

Revision as of 16:16, 7 January 2023

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:<math>\operatorname{li}(x)-\pi(x) = O(\sqrt{x}\log x)</math>

In fact, the [[Riemann hypothesis]] is equivalent to the fact 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 this happens]] is somewhere between 10¹⁹ and 1.4×10³¹⁶.