Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a non-elementary function defined for all positive real numbers ${\displaystyle x\neq 1}$ by the definite integral:

${\displaystyle {\rm {li}}(x)=\int _{0}^{x}{\frac {dt}{\ln(t)}}\;.}$

Here, ln denotes the natural logarithm. The function 1/ln (t) has a singularity at t = 1, and the integral for x > 1 has to be interpreted as a Cauchy principal value:

${\displaystyle {\rm {li}}(x)=\lim _{\varepsilon \to 0}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln(t)}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln(t)}}\right)\;.}$

Sometimes instead of li the offset logarithmic integral is used, defined as ${\displaystyle {\rm {Li}}(x)={\rm {li}}(x)-{\rm {li}}(2)}$. This is often used in number theoretic applications. Neither function should be confused with the logarithmic integral whose definition is

${\displaystyle \int _{-\infty }^{\infty }{\frac {M(t)}{1+t^{2}}}dt}$.

The growth behavior of this function for x → ∞ is

${\displaystyle {\rm {li}}(x)=\Theta \left({x \over \ln(x)}\right)\;.}$

(see big O notation).

The logarithmic integral finds application in many areas, in particular it is used is in estimates of prime number densities, such as the prime number theorem:

${\displaystyle \pi (x)\sim {\hbox{li}}(x)\sim {\hbox{Li}}(x)}$

where π(x) denotes the number of primes smaller than or equal to x.

The function li(x) is related to the exponential integral Ei(x) via the equation

${\displaystyle {\hbox{li}}(x)={\hbox{Ei}}(\ln(x))\quad {\hbox{for all positive real }}x\neq 1.}$

This leads to series expansions of li(x), for instance:

${\displaystyle {\rm {li}}(e^{u})=\gamma +\ln u+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\rm {for}}\;u\neq 0\;,}$

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 ...; this number is known as the Ramanujan-Soldner constant.

See also

• Jørgen Pedersen Gram
• Ramanujan-Soldner constant

References

• Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)