# Definition:Logarithmic Integral/Eulerian

## Definition

The **Eulerian logarithmic integral** is defined as:

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

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.

## Sources

- 1983: François Le Lionnais and Jean Brette:
*Les Nombres Remarquables*... (previous) ... (next): $0,5$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.16$: The Sequence of Primes