Definition:Logarithmic Integral/Eulerian

Definition

The Eulerian logarithmic integral is defined as:

$\ds \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$

The logarithmic integral and the Eulerian logarithmic integral are not consistently denoted in the literature (some sources use $\map {\operatorname {li} } x$ to indicate the Eulerian version, for example).

It is therefore important to take care which is being referred to at any point.

The Eulerian logarithmic integral is also known as the offset logarithmic integral.

This entry was named for Leonhard Paul Euler.

The Eulerian logarithmic integral was conjectured by Carl Friedrich Gauss when he was $14$ or $15$ to be a good approximation for the prime-counting function.

Hence the first statement of this particular form of the Prime Number Theorem.