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.

1 Integral representation
2 Offset logarithmic integral
3 Series representation
4 Special values
5 Asymptotic expansion
6 Infinite logarithmic integral
7 Number theoretic significance
8 See also
9 References

Integral representation

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

${\rm li} (x) = \int_0^x \frac{dt}{\ln t}. \;$

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:

${\rm li} (x) = \lim_{\varepsilon \to 0%2B} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} %2B \int_{1%2B\varepsilon}^x \frac{dt}{\ln t} \right). \;$

Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as

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

${\rm Li} (x) = \int_2^x \frac{dt}{\ln t} \,$

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

This function is a very good approximation to the number of prime numbers less than x.

Series representation

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

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

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

${\rm li} (e^u) = \hbox{Ei}(u) = \gamma %2B \ln u %2B \sum_{n=1}^\infty {u^{n}\over n \cdot n!} \quad \text{ for } u \ne 0 \; ,$

where γ ≈ 0.57721 56649 01532 ... is the Euler–Mascheroni gamma constant. A more rapidly convergent series due to Ramanujan ^[1] is

${\rm li} (x) = \gamma %2B \ln \ln x %2B \sqrt{x} \sum_{n=1}^\infty \frac{ (-1)^{n-1} (\ln x)^n} {n! \, 2^{n-1}} \sum_{k=0}^{\lfloor (n-1)/2 \rfloor} \frac{1}{2k%2B1} .$

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 $-(\Gamma\left(0,-\ln 2\right) %2B i\,\pi)$ where $\Gamma\left(a,x\right)$ 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

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

where $O$ is the big O notation. The full asymptotic expansion is

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

$\frac{{\rm li} (x)}{x/\ln x} \sim 1 %2B \frac{1}{\ln x} %2B \frac{2}{(\ln x)^2} %2B \frac{6}{(\ln x)^3} %2B \cdots.$

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

$\int_{-\infty}^\infty \frac{M(t)}{1%2Bt^2}dt$

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:

$\pi(x)\sim\operatorname{li}(x)$