Logarithmic Integral is Asymptotic to Prime-Counting Function

Theorem

Let $x \in \R$ be a real number such that $x > 2$.

Let $\map \li x$ denote the logarithmic integral of $x$.

Let $\map \pi x$ denote the prime-counting function of $x$.

Then $\map \li x$ is asymptotically equal to $\map \pi x$.

Proof

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Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithmic integral