Logarithmic integral

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The logarithmic integral is defined as either ("American" definition starting at 0, for which there is a singularity at $x$ = 1)

$\operatorname {li} (x):=\lim _{\eta \to 0^{+}}\left(\int _{0}^{1-\eta }+\int _{1+\eta }^{x}\right){\frac {du}{\log u}},$

or ("European" definition starting at 2)

$\operatorname {li} (x):=\int _{2}^{x}{\frac {du}{\log u}},$

where $\operatorname {li} (2)$ = 0 instead of 1.04516378011749278484458888919...

A069284 Decimal expansion of $Li(2)=\gamma +\log(\log(2))+\sum _{k=1}^{\infty }{\frac {\log(2)^{k}}{k!\,k}}$. (For "American" definition starting at 0.) ($\gamma$ being the Euler-Mascheroni constant.)

{1, 0, 4, 5, 1, 6, 3, 7, 8, 0, 1, 1, 7, 4, 9, 2, 7, 8, 4, 8, 4, 4, 5, 8, 8, 8, 8, 9, 1, 9, 4, 6, 1, 3, 1, 3, 6, 5, 2, 2, 6, 1, 5, 5, 7, 8, 1, 5, 1, 2, 0, 1, 5, 7, 5, 8, 3, 2, 9, 0, 9, 1, 4, 4, 0, 7, 5, 0, 1, 3, 2, 0, 5, 2, ...}

The prime number theorem states that the prime counting function is asymptotic to the logarithmic integral

$\pi (x)\sim \operatorname {li} (x).$

See also

• Skewes number

External links

• Weisstein, Eric W., Logarithmic Integral, from MathWorld—A Wolfram Web Resource.