# Logarithmic integral function

In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the Siegel-Walfisz theorem it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value ${\displaystyle x}$.

## Integral representation

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

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

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,

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

## Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as

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

or, integrally represented

${\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}\,}$

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

## Special values

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... ; this number is known as the Ramanujan–Soldner constant.

−Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151...

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

## Series representation

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

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

which is valid for x > 0. This identity provides a series representation of li(x) as

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

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

${\displaystyle \operatorname {li} (x)=\gamma +\ln \ln x+{\sqrt {x}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}(\ln x)^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}.}$

## Asymptotic expansion

The asymptotic behavior for x  ∞ is

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

where ${\displaystyle O}$ is the big O notation. The full asymptotic expansion is

${\displaystyle \operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}}$

or

${\displaystyle {\frac {\operatorname {li} (x)}{x/\ln x}}\sim 1+{\frac {1}{\ln x}}+{\frac {2}{(\ln x)^{2}}}+{\frac {6}{(\ln x)^{3}}}+\cdots .}$

This gives the following more accurate asymptotic behaviour:

${\displaystyle \operatorname {li} (x)-{x \over \ln x}=O\left({x \over \ln ^{2}x}\right)\;.}$

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.

This implies e.g. that we can bracket li as:

${\displaystyle 1+{\frac {1}{\ln x}}<\operatorname {li} (x){\frac {\ln x}{x}}<1+{\frac {1}{\ln x}}+{\frac {3}{(\ln x)^{2}}}}$

For all ${\displaystyle \ln x\geq 11}$.

## 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:

${\displaystyle \pi (x)\sim \operatorname {Li} (x)}$

where ${\displaystyle \pi (x)}$ denotes the number of primes smaller than or equal to ${\displaystyle x}$.

Assuming the Riemann hypothesis, we get the even stronger:[2]

${\displaystyle \operatorname {Li} (x)-\pi (x)=O({\sqrt {x}}\log x)}$