Definition:Logarithmic Integral/Eulerian
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Definition
The Eulerian logarithmic integral 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.
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