Eulerian 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 Eulerian 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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Examples

Example: $10^6$

\(\ds \map \Li {10^6}\)

\(\approx\)

\(\ds 78 \, 628\)

\(\ds \map \pi {10^6}\)

\(=\)

\(\ds 78 \, 498\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\map \Li {10^6} } {\map \pi {10^6} }\)

\(\approx\)

\(\ds 1 \cdotp 00166\)

Sources

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