Logarithmic integral function

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Plot of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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 Template:Mvar.

File:Logarithmic integral function.svg

Logarithmic integral function plot

Integral representation

The logarithmic integral has an integral representation defined for all positive real numbers Template:Mvar ≠ 1 by the definite integral

<math> \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t}. </math>

Here, Template:Math denotes the natural logarithm. The function Template:Math has a singularity at Template:Math, and the integral for Template:Math is interpreted as a Cauchy principal value,

<math> \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).</math>

Offset logarithmic integral

The offset logarithmic integral or Eulerian logarithmic integral is defined as

<math> \operatorname{Li}(x) = \int_2^x \frac{dt}{\ln t} = \operatorname{li}(x) - \operatorname{li}(2). </math>

As such, the integral representation has the advantage of avoiding the singularity in the domain of integration.

Equivalently,

<math> \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t} = \operatorname{Li}(x) + \operatorname{li}(2). </math>

Special values

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... Template:OEIS2C; this number is known as the Ramanujan–Soldner constant.

<math>\operatorname{li}(\text{Li}^{-1}(0)) = \text{li}(2)</math> ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... Template:OEIS2C

This is <math>-(\Gamma(0,-\ln 2) + i\,\pi)</math> where <math>\Gamma(a,x)</math> 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

<math>\operatorname{li}(x)=\hbox{Ei}(\ln x) ,</math>

which is valid for x > 0. This identity provides a series representation of li(x) as

<math> \operatorname{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 \, , </math> where γ ≈ 0.57721 56649 01532 ... Template:OEIS2C is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan <ref>Template:MathWorld</ref> is

<math>

\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).

</math>

Asymptotic expansion

The asymptotic behavior for <math>x\to\infty</math> is

<math> \operatorname{li}(x) = O \left( \frac{x }{\ln x} \right) . </math>

where <math>O</math> is the big O notation. The full asymptotic expansion is

<math> \operatorname{li}(x) \sim \frac{x}{\ln x} \sum_{k=0}^\infty \frac{k!}{(\ln x)^k} </math>

or

<math> \frac{\operatorname{li}(x)}{x/\ln x} \sim 1 + \frac{1}{\ln x} + \frac{2}{(\ln x)^2} + \frac{6}{(\ln x)^3} + \cdots. </math>

This gives the following more accurate asymptotic behaviour:

<math> \operatorname{li}(x) - \frac{x}{ \ln x} = O \left( \frac{x}{(\ln x)^2} \right) . </math>

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:

for all <math>\ln x \ge 11</math>.

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:

<math>\pi(x)\sim\operatorname{li}(x)</math>

where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>.

Assuming the Riemann hypothesis, we get the even stronger:<ref>Abramowitz and Stegun, p. 230, 5.1.20</ref>

<math>|\operatorname{li}(x)-\pi(x)| = O(\sqrt{x}\log x)</math>

In fact, the Riemann hypothesis is equivalent to the statement that:

<math>|\operatorname{li}(x)-\pi(x)| = O(x^{1/2+a})</math> for any <math>a>0</math>.

For small <math>x</math>, <math>\operatorname{li}(x)>\pi(x)</math> but the difference changes sign an infinite number of times as <math>x</math> increases, and the first time that this happens is somewhere between 10¹⁹ and Template:Val.

References

Template:Reflist

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