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More clarifications on definition range near point 1 |
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Definite [[integral]] defined as:
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is a non-elemental [[function]] called '''logarithmic integral''' or '''integral logarithm'''. For ''x'' > 1 in a point ''t''=1 this integral diverges
:li(e<sup>''u''</sup>) = γ + ln ''u'' + ''u'' + ''u''<sup>2</sup>/2 · 2! + ''u''<sup>3</sup>/3 · 3! + ''u''<sup>4</sup>/4 · 4! - ...,
where γ ≈ 0.57721 56649 01532 ... is [[Leonhard Euler|Euler-Mascheroni's constant]]. The logarithmic integral also obeys next identity:
:li(x<sup>''u''</sup>) = γ + ln ln ''u'' - ln ''u'' + <font size="+
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 with no Cauchy's principal value so that Li(2) = 0 and the estimation for ''prime counting function'' π(''n'') is:
: π(''n'') ~ '''Li(n)''' ≡ <font size="+
:Li(''x'') = li(''x'') - li(2) ≈ li(''x'') - 1.
If we want to avoid singular value in point 1 we sometimes take a constant, denoted with '''c''' or '''μ'''>1 in a way that is:
:Cpv <font size="+1">∫</font><sub>0</sub><sup>μ</sup> 1/ln ''t'' d''t''.
Thus we can rewrite Cpv ∫<sub>''0''</sub><sup>''x''</sup> 1/ln ''t'' d''t'' by ∫<sub>''μ''</sub><sup>''x''</sup> 1/ln ''t'' d''t'' in ''x''>1. This method was first used by [[Srinivasa Aaiyangar Ramanujan]] and this constant μ ≈ 1.45136 92348 ... is now called ''Ramanujan-Soldner constant'' or ''Soldner constant''. It represents a zero of an equation li(''x'')=0. Ramanujan calculated μ ≈ 1.45136 3380 ... This constant appears in the following form of prime numbers theorem:
: π(''n'') = <font size="+1">∑</font><sub>m=1</sub><sup>''∞''</sup> μ(''m'')/''m'' <font size="+1">∫</font><sub>μ</sub><sup>n</sup> 1/ ln ''t'' d''t'',
where μ(''m'') is [[Moebius function|Möbius function]].
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