Definition:Logarithmic Integral/Eulerian

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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.