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Logarithmic integral function - Wikipedia

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"Li(x)" redirects here. For the polylogarithm denoted by Lis(z), see Polylogarithm.

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
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 {\displaystyle x}.

Logarithmic integral function plot

Integral representation

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

{\displaystyle \operatorname {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 is interpreted as a Cauchy principal value,

{\displaystyle \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).}

Offset logarithmic integral

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The offset logarithmic integral or Eulerian logarithmic integral is defined as

{\displaystyle \operatorname {Li} (x)=\int _{2}^{x}{\frac {dt}{\ln t}}=\operatorname {li} (x)-\operatorname {li} (2).}

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

Equivalently,

{\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}}=\operatorname {Li} (x)+\operatorname {li} (2).}

The function li(x) has a single positive zero; it occurs at x ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930... OEIS: A070769; this number is known as the Ramanujan–Soldner constant.

{\displaystyle \operatorname {li} ({\text{Li}}^{-1}(0))={\text{li}}(2)} ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... OEIS: A069284

This is {\displaystyle -(\Gamma (0,-\ln 2)+i\,\pi )} where {\displaystyle \Gamma (a,x)} is the incomplete gamma function. It must be understood as the Cauchy principal value of the function.

Series representation

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The function li(x) is related to the exponential integral Ei(x) via the equation

{\displaystyle \operatorname {li} (x)={\hbox{Ei}}(\ln x),}

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

{\displaystyle \operatorname {li} (e^{u})={\hbox{Ei}}(u)=\gamma +\ln |u|+\sum _{n=1}^{\infty }{u^{n} \over n\cdot n!}\quad {\text{ for }}u\neq 0\,,}

where γ ≈ 0.57721 56649 01532 ... OEIS: A001620 is the Euler–Mascheroni constant. A more rapidly convergent series by Ramanujan [1] is

{\displaystyle \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).}

Asymptotic expansion

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The asymptotic behavior for x → ∞ is

{\displaystyle \operatorname {li} (x)=O\left({\frac {x}{\ln x}}\right).}

where {\displaystyle O} is the big O notation. The full asymptotic expansion is

{\displaystyle \operatorname {li} (x)\sim {\frac {x}{\ln x}}\sum _{k=0}^{\infty }{\frac {k!}{(\ln x)^{k}}}}

or

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

This gives the following more accurate asymptotic behaviour:

{\displaystyle \operatorname {li} (x)-{\frac {x}{\ln x}}=O\left({\frac {x}{(\ln x)^{2}}}\right).}

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:

{\displaystyle 1+{\frac {1}{\ln x}}<\operatorname {li} (x){\frac {\ln x}{x}}<1+{\frac {1}{\ln x}}+{\frac {3}{(\ln x)^{2}}}}

for all {\displaystyle \ln x\geq 11}.

Number theoretic significance

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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:

{\displaystyle \pi (x)\sim \operatorname {li} (x)}

where {\displaystyle \pi (x)} denotes the number of primes smaller than or equal to {\displaystyle x}.

Assuming the Riemann hypothesis, we get the even stronger:[2]

{\displaystyle |\operatorname {li} (x)-\pi (x)|=O({\sqrt {x}}\log x)}

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

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

For small {\displaystyle x}, {\displaystyle \operatorname {li} (x)>\pi (x)} but the difference changes sign an infinite number of times as {\displaystyle x} increases, and the first time that this happens is somewhere between 1019 and 1.4×10316.

  1. ^ Weisstein, Eric W. "Logarithmic Integral". MathWorld.
  2. ^ Abramowitz and Stegun, p. 230, 5.1.20