Logarithmic integral function: Difference between revisions

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Definite [[integral]] defined as:

: '''li(''x'')''' &equiv; <font size="+2">&int;</font>_''0''^''x'' 1/ln ''t'' d''t''

is a non-elemental [[function]] called '''logarithmic integral''' or '''integral logarithm''' and denoted with '''li(''x'')''' or '''Li(''x'')'''. For ''x'' > 1 in a point ''t''=1 this integral diverges, in this case we use for Lili(''x'') the main value of unessential integral. LogarithmicThis integral with the main value of nondefinite integral comesis in a varietyconnection ofwith formulas''integral concerningexponential thefunction'' densityor of''exponential [[primeintegral'' number|primes]]such inas [[numberthat theory]]li(''x'') and= speciallyEi in(ln [[prime''x''). numberIf theorem|primewe numbers theorem]], where for example the estimation forsubstitute ''prime counting functionx'' &pi;(with e^''nu''), iswe get a series:

: &pi;li(e^{''nu'') ~ Li(n}) = &intgamma;<sub> + ln ''2u'' + ''u'' + ''u''</subsup>2</sup>/2 &middot; 2! + ''nu''³ 1/3 ln&middot; ''t''3! + d''tu''.⁴/4 &middot; 4! - ...,

where &gamma; &#8776; 0.57721 56649 01532 is [[Leonhard Euler|Euler-Mascheroni's constant]]. The logarithmic integral also obeys next identity:

This integral is in a connection with ''integral exponential function'' such as that li(''x'') = Ei (ln ''x''). If we substitute ''x'' with e^''u'', we get a series:

:li(ex^''u'') = &gamma; + ln ln ''u'' +- ln ''u'' + ''u''<sup>font size="+2"> &sum;</supfont><sub>n=1</2 sub>^{&middotinfin;} 2! +(ln ''u'')^3''n''/3''n'' &middot; 3''n''! + ''um''^4''n''/4 &middot; 4! - ...,

Logarithmic integral with the main value of nondefinite integral comes in a variety of formulas concerning the density of [[prime number|primes]] in [[number theory]] and specially in [[prime number theorem|prime numbers theorem]], where for example logarithmic integral is defined so that Li(2) = 0 and the estimation for ''prime counting function'' &pi;(''n'') is:

: &pi;(''n'') ~ '''Li(n)''' &equiv; <font size="+2">&int;</font>_''2''^''n'' 1/ ln ''t'' d''t''.

:Li(''n'') = li(''x'') - li(2) &#8776; li(''x'') - 1.04516 = Ei (ln ''x'')

where &gamma; &#8776; 0.57721 56649 01532 is [[Leonhard Euler|Euler-Mascheroni's constant]].