Logarithmic Integral Calculator

The Logarithmic Integral function li(x) is defined by the integral

li(x) = ∫^x₀  dt/ln(t)

The antiderivative of the function f(t) = 1/ln(t) cannot be written explicitly in terms of elementary functions, thus the logarithmic integral function cannot be evaluated in terms of other elementary functions such as ln(x) or e^x. Because the function 1/ln(t) has a singularity at t = 0, an alternative form of the logarithmic integral is sometimes used:

Li(x) = ∫^x₂  dt/ln(t)  =  li(x) - li(2)

The logarithmic integral is important in number theory since the number of primes less than or equal to n is roughly Li(n).

Other Properties of li(x)

The logarithmic integral can be expressed as an infinite series:

li(x) = γ + ln(ln(x)) + Σ^∞ _n=1   ln(x)ⁿ / (n⋅n!)

where γ is the Euler-Mascheroni constant 0.57721566490153286...

For large values of x, li(x) can be evaluated with the asymptotic formula

li(x) ∼ x/ln(x) + x/ln(x)² + 2x/ln(x)³ + 6x/ln(x)⁴ + 24x/ln(x)⁵

At x = 1.45136923488338105... the function li(x) is equal to 0.

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