Definitions

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.

Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers xne 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 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.

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 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 -(Gammaleft(0,-ln 2right) + i,pi) where Gammaleft(a,xright) 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) = mathcal{O} left({xover 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.

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.

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)simhbox{li}(x)simhbox{Li}(x)

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

See also

References

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