Definition:Logarithmic Integral/Eulerian
Definition
Let $x \in \R$ be a real number such that $x > 2$.
The Eulerian logarithmic integral of $x$ is defined as:
- $\ds \map \Li x = \int_2^x \frac {\d t} {\ln t}$
where $\ln$ denotes the natural logarithm function.
Also known as
The Eulerian logarithmic integral is also known as the offset logarithmic integral.
Some sources refer to the Eulerian logarithmic integral just as the logarithmic integral, but this can cause confusion.
Warning
The logarithmic integral and the Eulerian logarithmic integral are not consistently denoted in the literature (some sources use $\map \li x$ to indicate the Eulerian version, for example).
It is therefore important to take care which is being referred to at any point.
Also see
- Results about the Eulerian logarithmic integral can be found here.
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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logarithmic integral