Logarithmic integral function: Difference between revisions

Revision as of 08:14, 20 September 2006

Revision as of 08:41, 20 September 2006

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\quad {\rm for} \; u \ne 0 \; , </math>

where &gamma;γ &asymp;≈ 0.57721 56649 01532 ... is the [[Euler-Mascheroni gamma constant]].

==Special values==

The function li(''x'') has a single positive zero; it occurs at ''x'' &asymp;≈ 1.45136 92348 ...; this number is known as the [[Ramanujan-Soldner constant]].

One has li(2) &asymp;≈ 1.04516 37801 17492 ...

==Asymptotic expansion==

The asymptotic behavior for ''x'' &rarr;→ &infin;∞ is

:<math> {\rm li} (x) = \mathcal{O} \left( {x\over \ln (x)} \right) \; . </math>

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:<math>\pi(x)\sim\hbox{li}(x)\sim\hbox{Li}(x)</math>

where &pi;π(''x'') denotes the number of primes smaller than or equal to ''x''.

== See also ==