Logarithmic integral function: Difference between revisions

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Revision as of 16:49, 2 February 2017

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This function is a very good approximation to the number of prime numbers less than x.

==Special values==

The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 ... {{OEIS2C|A070769}}; this number is known as the [[Ramanujan–Soldner constant]].

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151… {{OEIS2C|A069284}}

This is <math>-(\Gamma\left(0,-\ln 2\right) + i\,\pi)</math> where <math>\Gamma\left(a,x\right)</math> is the [[incomplete gamma function]]. It must be understood as the [[Cauchy principal value]] of the function.

==Series representation==

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Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 126–131, 1994.

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==Special values==

The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 ... {{OEIS2C|A070769}}; this number is known as the [[Ramanujan–Soldner constant]].

li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151… {{OEIS2C|A069284}}

This is <math>-(\Gamma\left(0,-\ln 2\right) + i\,\pi)</math> where <math>\Gamma\left(a,x\right)</math> is the [[incomplete gamma function]]. It must be understood as the [[Cauchy principal value]] of the function.

==Asymptotic expansion==