Definitions

# Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It occurs in problems of physics and has number theoretic significance, occurring in the prime number theorem as an estimate of the number of prime numbers less than a given value.

## Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers $xne 1$ by the definite integral:

$\left\{rm li\right\} \left(x\right) = int_\left\{0\right\}^\left\{x\right\} frac\left\{dt\right\}\left\{ln \left(t\right)\right\} ; .$

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:

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

## Offset logarithmic integral

The offset logarithmic integral or Eulerean logarithmic integral is defined as

$\left\{rm Li\right\}\left(x\right) = \left\{rm li\right\}\left(x\right) - \left\{rm li\right\}\left(2\right) ,$

or

$\left\{rm Li\right\} \left(x\right) = int_\left\{2\right\}^\left\{x\right\} frac\left\{dt\right\}\left\{ln t\right\} ,$

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

## Series representation

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

$hbox\left\{li\right\}\left(x\right)=hbox\left\{Ei\right\}\left(ln\left(x\right)\right) , ,!$

which is valid for $x > 1$. This identity provides a series representation of li(x) as

$\left\{rm li\right\} \left(e^\left\{u\right\}\right) = hbox\left\{Ei\right\}\left(u\right) =$
gamma + ln u + sum_{n=1}^{infty} {u^{n}over n cdot n!} quad {rm for} ; u ne 0 ; ,

where γ ≈ 0.57721 56649 01532 ... is the Euler-Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan is


{rm li} (x) =
`gamma`
`+ ln ln x`
+ sqrt{x} sum_{n=1}^{infty} frac{ (-1)^{n-1} (ln x)^n} {n! , 2^{n-1}} sum_{k=0}^{lfloor (n-1)/2 rfloor} frac{1}{2k+1} .

## Special values

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.

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…

This is $-\left(Gammaleft\left(0,-ln 2right\right) + i,pi\right)$ where $Gammaleft\left(a,xright\right)$ is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

## Asymptotic expansion

The asymptotic behavior for x → ∞ is

$\left\{rm li\right\} \left(x\right) = mathcal\left\{O\right\} left\left(\left\{xover ln \left(x\right)\right\} right\right) ; .$

where $mathcal\left\{O\right\}$ refers to big O notation. The full asymptotic expansion is

$\left\{rm li\right\} \left(x\right) = frac\left\{x\right\}\left\{ln x\right\} sum_\left\{k=0\right\}^\left\{infty\right\} frac\left\{k!\right\}\left\{\left(ln x\right)^k\right\}$

or

$frac\left\{\left\{rm li\right\} \left(x\right)\right\}\left\{x/ln x\right\} = 1 + frac\left\{1\right\}\left\{ln x\right\} + frac\left\{2\right\}\left\{\left(ln x\right)^2\right\} + frac\left\{6\right\}\left\{\left(ln x\right)^3\right\} + cdots.$

Note that, as an asymptotic expansion, this series is not convergent: it is a reasonable approximation only if the series is truncated at a finite number of terms, and only large values of x are employed. This expansion follows directly from the asymptotic expansion for the exponential integral.

## Infinite logarithmic integral

$int_\left\{-infty\right\}^infty frac\left\{M\left(t\right)\right\}\left\{1+t^2\right\}dt$
and discussed in Paul Koosis, The Logarithmic Integral, volumes I and II, Cambridge University Press, second edition, 1998.

## Number theoretic significance

The logarithmic integral is important in number theory, appearing in estimates of the number of prime numbers less than a given value. For example, the prime number theorem states that:

$pi\left(x\right)simhbox\left\{li\right\}\left(x\right)simhbox\left\{Li\right\}\left(x\right)$

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