Logarithmic integral function: Difference between revisions

Revision as of 16:16, 7 January 2023

Revision as of 16:49, 8 January 2023

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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 factstatement that:

:<math>|\operatorname{li}(x)-\pi(x)| = O(x^{1/2+a})</math> for any <math>a>0</math>.