Logarithmic integral function


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.

Logarithmic integral function plot

Contents

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+} \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

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

or

 {\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 + \ln u + \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
 + \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 -(\Gamma\left(0,-\ln 2\right) + 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}

or

 \frac{{\rm li} (x)}{x/\ln x}  \sim  1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \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

[clarification needed]:\int_{-\infty}^\infty \frac{M(t)}{1+t^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)

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

See also

References

  1. ^ Weisstein, Eric W., "Logarithmic Integral" from MathWorld.

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