Logarithmic integral function

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See also logarithmic integral for other senses.

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

1 Integral representation
2 Offset logarithmic integral
3 Series representation
4 Special values
5 Asymptotic expansion
6 Number theoretic significance
7 See also
8 References

[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

Li(x) = li(x) - li(2)

${\rm Li} (x) = \int_{2}^{x} \frac{dt}{\ln t} \,$

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

li(x) = 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.

[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.

One has li(2) ≈ 1.04516 37801 17492 ...

[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

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

$\frac{{\rm li} (x)}{x/\ln x} = 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \cdots.$

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] 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

Jørgen Pedersen Gram

[edit] References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See Chapter 5)

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Categories: Special functions | Special hypergeometric functions