# 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 prime number 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 $x$.

## Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral
    $\operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}.$
Here, ln denotes the natural logarithm. The function 1/(ln t) has a singularity at t = 1, and the integral for x > 1 is interpreted as a Cauchy principal value,
    $\operatorname {li} (x)=\lim _{\varepsilon \to 0+}\left(\int _{0}^{1-\varepsilon }{\frac {dt}{\ln t}}+\int _{1+\varepsilon }^{x}{\frac {dt}{\ln t}}\right).$

## Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as
    $\operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}=\operatorname {li} (x)-\operatorname {li} (2).$
As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.
Equivalently,
    $\operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=\operatorname {Li} (x)+\operatorname {li} (2).$

## Special values

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEIS: A070769; this number is known as the Ramanujan–Soldner constant.
$\operatorname {li} ({\text{Li}}^{-1}(0))={\text{li}}(2)$ ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284
This is $-(\Gamma (0,-\ln 2)+i\,\pi )$ where $\Gamma (a,x)$ 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
    $\operatorname {li} (x)={\hbox{Ei}}(\ln x),$
which is valid for x > 0. This identity provides a series representation of li(x) as
    $\operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln |u|+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0\,,$
where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan is
    $\operatorname {li} (x)=\gamma +\ln |\ln x|+{\sqrt {x}}\sum _{n=1}^{\infty }\left({\frac {(-1)^{n-1}(\ln x)^{n}}{n!\,2^{n-1}}}\sum _{k=0}^{\lfloor (n-1)/2\rfloor }{\frac {1}{2k+1}}\right).$

## Asymptotic expansion

The asymptotic behavior for x → ∞ is
    $\operatorname {li} (x)=O\left({\frac {x}{\ln x}}\right).$
where $O$ is the big O notation. The full asymptotic expansion is
    $\operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}$
or
    ${\frac {\operatorname {li} (x)}{x/\ln x}}\sim 1+{\frac {1}{\ln x}}+{\frac {2}{(\ln x)^{2}}}+{\frac {6}{(\ln x)^{3}}}+\cdots .$
This gives the following more accurate asymptotic behaviour:
    $\operatorname {li} (x)-{\frac {x}{\ln x}}=O\left({\frac {x}{(\ln x)^{2}}}\right).$
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:
    $1+{\frac {1}{\ln x}}<\operatorname {li} (x){\frac {\ln x}{x}}<1+{\frac {1}{\ln x}}+{\frac {3}{(\ln x)^{2}}}$
for all $\ln x\geq 11$.

## 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:
    $\pi (x)\sim \operatorname {li} (x)$
where $\pi (x)$ denotes the number of primes smaller than or equal to $x$.
Assuming the Riemann hypothesis, we get the even stronger:
    $|\operatorname {li} (x)-\pi (x)|=O({\sqrt {x}}\log x)$
In fact, the Riemann hypothesis is equivalent to the statement that:
    $|\operatorname {li} (x)-\pi (x)|=O(x^{1/2+a})$ for any $a>0$.
For small $x$, $\operatorname {li} (x)>\pi (x)$ but the difference changes sign an infinite number of times as $x$ increases, and the first time that this happens is somewhere between 10$^{19}$ and 1.4×10$^{316}$.