Logarithmic integral function

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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
Logarithmic integral

Contents

[edit] 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) \; .

[edit] Offset logarithmic integral

The offset logarithmic integral or Eulerean 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.

[edit] 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 {\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[citation needed] 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} .

[edit] 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.

[edit] Asymptotic expansion

The asymptotic behavior for x → ∞ is

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

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

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

or

\frac{{\rm li} (x)}{x/\ln x} = 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.

[edit] Infinite logarithmic integral

\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.

[edit] 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\hbox{li}(x)\sim\hbox{Li}(x)

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

[edit] See also

[edit] References

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