Logarithmic integral function: Difference between revisions

Revision as of 20:39, 20 December 2005

Revision as of 18:41, 21 December 2005

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the:''See also [[logarithmic integral]] whosefor definitionother issenses.''

{{inuse}}

In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[function (mathematics)|non-elementary function]] defined for all positive [[real number]]s <math>x\ne1</math> by the [[integral|definite integral]]:

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Sometimes instead of li the [[offset logarithmic integral]] is used, defined as

<math>{\rm Li}(x) = {\rm li}(x) - {\rm li}(2)</math>. This is often used in number theoretic applications. Neither function should be confused with

the [[logarithmic integral]] whose definition is

:<math>\int_{-\infty}^\infty \frac{M(t)}{1+t^2}dt</math>.

The growth behavior of this function for ''x'' &rarr; &infin; is

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(see [[big O notation]]).

==Series representation==

The logarithmic integral finds application in many areas, in particular it is used is in estimates of [[prime number]] densities, such as the [[prime number theorem]]:

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

where &pi;(''x'') denotes the number of primes smaller than or equal to ''x''.

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

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where &gamma; &asymp; 0.57721 56649 01532 ... is the [[Euler-Mascheroni gamma constant]]. The function li(''x'') has a single positive zero; it occurs at ''x'' &asymp; 1.45136 92348 ...; this number is known as the [[Ramanujan-Soldner constant]].

==Number theoretic significance==

The logarithmic integral finds application in many areas, in particular it is used is in estimates of [[prime number]] densities, such as the [[prime number theorem]]:

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

where &pi;(''x'') denotes the number of primes smaller than or equal to ''x''.

== See also ==

* [[Jørgen Pedersen Gram]]

* [[Ramanujan-Soldner constant]]

== References ==