Logarithmic integral function: Difference between revisions

Revision as of 03:25, 28 January 2019

Revision as of 23:05, 6 June 2019

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==Special values==

The function li(''x'') has a single positive zero; it occurs at ''x'' ≈ 1.45136 92348 83381 05028 39684 85892 02744 94930…94930... {{OEIS2C|A070769}}; this number is known as the [[Ramanujan–Soldner constant]].

−Li(0) = li(2) ≈ 1.045163 780117 492784 844588 889194 613136 522615 578151…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.

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:<math> \operatorname{li}(x) - {x\over \ln x} = O \left( {x\over \ln^2 x} \right) \; . </math>

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]].

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where <math>\pi(x)</math> denotes the number of primes smaller than or equal to <math>x</math>.

Assuming Riemann’sRiemann's hypothesis, we get the even stronger:<ref>Abramowitz and Stegun, p.&nbsp;230, 5.1.20</ref>:

:<math>\operatorname{Li}(x)-\pi(x) = O(\sqrt{x}\log x)</math>