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: ∫<sub>''0''</sub><sup>''x''</sup> 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 Li(''x'') the main value of unessential integral. 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|
: π(''n'') ~ Li(n) = ∫<sub>''2''</sub><sup>''n''</sup> 1/ ln ''t'' d''t''.
This integral is in a connection with ''integral exponential function'' such as that li(''x'') = Ei (ln ''x''). If we substitute ''x'' with e<sup>''u''</sup>, we get a serie:
: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]].
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