# Definition:Logarithmic Integral/Eulerian

## Definition

The Eulerian logarithmic integral is defined as:

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

where $\ln$ denotes the natural logarithm function.

## Also defined as

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.

## Also known as

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

## Source of Name

This entry was named for Leonhard Paul Euler.

## Historical Note

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.