Logarithmic integral function: Difference between revisions

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{{Mergeto|logarithmic integral|date=May 2007}}

:''See also [[logarithmic integral]] for other senses.''

In [[mathematics]], the '''logarithmic integral function''' or '''integral logarithm''' li(''x'') is a [[special function]]. It occurs in problems of [[physics]] and has [[number theory|number theoretic]] significance, occurring in the [[prime number theorem]] as an [[estimate]] of the number of [[prime number]]s less than a given value.

[[Image:Logarithmic integral.svg|thumb|right|Logarithmic integral]]

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Note that, 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]].

{{Mergeto|== Infinite logarithmic integral|date=May 2007}}==

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

and discussed in Paul Koosis, ''The Logarithmic Integral'',

volumes I and II, Cambridge University Press, second edition, 1998.

==Number theoretic significance==