Logarithmic integral function: Difference between revisions

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{{Redirect|Li(x)|the polylogarithm denoted by Li_''s''(''z'')|Polylogarithm}}

{{Use American English|date = January 2019}}

[[File: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.svg|alt=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|thumb|Plot of the absolute value of the logarithmic integral function li(z) in the complex plane from -2-2i to 2+2i with colors createdshowing withthe Mathematicaargument 13.1(the functionangle ComplexPlot3Daround the complex plane)]]

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 theory|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 <math>{{mvar|x</math>}}. [[Image:Logarithmic integral function.svg|thumb|right|300px|Logarithmic integral function plot]]

== Integral representation ==

The logarithmic integral has an integral representation defined for all positive [[real number]]s {{mvar|x}}&nbsp;≠&nbsp;1 by the [[integral|definite integral]]

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

Here, {{math|ln}} denotes the [[natural logarithm]]. The function {{math|1/(ln ''t'')}} has a [[mathematical singularity|singularity]] at {{math|1=''t'' = 1}}, and the integral for {{math|''x'' > 1}} 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>

However, the logarithmic integral can also be taken to be a [[meromorphic]] complex-valued function in the complex ___domain. In this case it is multi-valued with branch points at 0 and 1, and the values between 0 and 1 defined by the above integral are not compatible with the values beyond 1. The complex function is shown in the figure above. The values on the real axis beyond 1 are the same as defined above, but the values between 0 and 1 are offset by iπ so that the absolute value at 0 is π rather than zero. The complex function is also defined (but multi-valued) for numbers with negative real part, but on the negative real axis the values are not real.

==Offset logarithmic integral==

The '''offset logarithmic integral''' or '''Eulerian logarithmic integral''' is defined as

== Offset logarithmic integral ==

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

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 ==

:<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... {{OEIS2C|A070769}}; this number is known as the [[Ramanujan–Soldner constant]].

<math>\textoperatorname{li}(\text{Li}^{-1}(0)) = \text{li}(2)</math> ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151... {{OEIS2C|A069284}}

This is <math>-(\Gamma\left(0,-\ln 2\right) + i\,\pi)</math> where <math>\Gamma\left(a,x\right)</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>\hboxoperatorname{li}(x)=\hbox{Ei}(\ln x) , \,\!</math>

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

which is valid for ''x''&nbsp;>&nbsp;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 ... {{OEIS2C|id=A001620}} is the [[Euler–Mascheroni constant]]. AFor morethe rapidlycomplex convergentfunction seriesthe by [[Srinivasa Ramanujan|Ramanujan]] <ref>{{MathWorld | urlname=LogarithmicIntegral | title=Logarithmic Integral}}</ref>formula is

: <math>| \operatorname{li}(xe^u)-\pi(x)| = O(\sqrthbox{xEi}\log(u) = x)</math>

:<math>

\gamma + \ln u + \sum_{n=1}^\infty {u^{n}\over n \cdot n!}

\quad \text{ for } u \ne 0 \, , </math>

(without taking the absolute value of u).

A more rapidly convergent series by [[Srinivasa Ramanujan|Ramanujan]] <ref>{{MathWorld | urlname=LogarithmicIntegral | title=Logarithmic Integral}}</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>

<!-- cribbed from Mathworld, which cites

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126–131, 1994.

-->

Again, for the meromorphic complex function the term <math>\ln|\ln u|</math> must be replaced by <math>\ln\ln u.</math>

== Asymptotic expansion ==

The asymptotic behavior both for ''<math>x''&nbsp;→&nbsp;∞\to\infty</math> and for <math>x\to 0^+</math> is

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

:<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>

:<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>

:<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>

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

As an asymptotic expansion, this series is [[divergent series|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:

: <math> 1+\frac{1}{\ln x} < \operatorname{li}(x) \frac{\ln x}{x} < 1+\frac{1}{\ln x}+\frac{3}{(\ln x)^2} </math>

:<math> 1+\frac{1}{\ln x} < \operatorname{li}(x) \frac{\ln x}{x} < 1+\frac{1}{\ln x}+\frac{3}{(\ln x)^2} </math>

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 number]]s less than a given value. For example, the [[prime number theorem]] states that:

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

:<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.&nbsp;230, 5.1.20</ref>

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

:<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 [[Skewes's number|first time that this happens]] is somewhere between 10¹⁹ and {{val|1.4×10^4|e=316}}.

:<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 [[Skewes's number|first time this happens]] is somewhere between 10¹⁹ and 1.4×10³¹⁶.

== See also ==

Line 104 ⟶ 98:

== References ==

{{Reflistreflist}}

* {{AS ref|5|228}}

* {{dlmf|id=6|title=Exponential, Logarithmic, Sine, and Cosine Integrals|first=N. M. |last=Temme}}

{{Nonelementary Integral}}