There are two common definitions that differ by a constant. For one of the most important applications - as an asymptotic comparison variable for the prime number function in the prime number theorem - the difference between the two definitions does not matter.

One definition in the field is
${\ displaystyle x \ geq 0}$

it must because of the singularity in for a limit to be defined ( Cauchy principal value ):
${\ displaystyle \ operatorname {li}}$${\ displaystyle x = 1}$${\ displaystyle x> 1}$

It applies with the integral exponential function , from which one obtains the series representation
${\ displaystyle \ operatorname {li} (x) = \ operatorname {Ei} (\ ln x)}$${\ displaystyle \ operatorname {egg}}$

where for, because of the singularity, Cauchy's principal value must be used.
We also have for${\ displaystyle x> 1}$${\ displaystyle t = 1 / x}$ ${\ displaystyle x \ geq 0, x \ neq 1}$

for one obtains
In the limiting case is${\ displaystyle p = 1}$${\ displaystyle \ textstyle \ int _ {0} ^ {1} \ operatorname {li} (t) \, \ mathrm {d} t = - \ ln 2.}$ ${\ displaystyle p = 0}$${\ displaystyle \ textstyle \ int _ {0} ^ {1} \ operatorname {li} (t) \, t ^ {- 1} \, \ mathrm {d} t = -1.}$

Another formula is ${\ displaystyle \ textstyle \ int _ {0} ^ {1} \ operatorname {li} (t ^ {- 1}) \, t \, \ mathrm {d} t = \ textstyle \ int _ {1} ^ { \ infty} \ operatorname {li} (t) \, t ^ {- 3} \, \ mathrm {d} t = 0.}$

The Golomb-Dickman constant (sequence A084945 in OEIS ) occurs in the theory of random permutations when estimating the length of the longest cycle of a permutation and in number theory when estimating the size of the largest prime factor of a number.
${\ displaystyle \ lambda = \ textstyle \ int _ {0} ^ {1} \ mathrm {e} ^ {\ operatorname {li} (x)} \ mathrm {d} x = 0 {,} 62432 \; 99885 \ ; 43550 \; 87099 \ ldots}$

Asymptotic behavior

Function graph of in the range between 1 and 10 ^{13}${\ displaystyle \ operatorname {li} (x)}$^{}

For big ones it can
${\ displaystyle x}$${\ displaystyle \ operatorname {li} (x)}$

approximate . The series is an asymptotic development ; it does not converge , but rather approaches the true value and then moves away again. The best approximation is achieved after roughly terms, then the summands become larger due to the increasing effect of the factorial .
${\ displaystyle \ ln x}$