cp's OEIS Frontend

From Mats Granvik, Jun 14 2013: (Start)

The logarithmic integral li(x) = exponential integral Ei(log(x)).

The generating function for tau A000005, the number of divisors of n is: Sum_{n >= 1} a(n) x^n = Sum_{k > 0} x^k/(1 - x^k). Another way to write the generating function for tau A000005 is Sum_{n>=1} A000005(n) x^n = Sum_{a=1..Infinity} Sum_{b>=1} x^(a*b).

If we instead think of the integral with the same form, evaluate at x = exp(1) = 2.7182818284... = A001113 and set the integration limits to zero and sqrt(log(n)), we get for n >= 0:

Logarithmic integral li(n) = Integral_{a = 0..sqrt(log(n))} Integral_{b=0..sqrt(log(n))} exp(1)^(a*b) + EulerGamma + log(log(n)). (End)

li(2)-1 is the minimum [known to date, for n>1] of |li(n) - PrimePi(n)|. - Jean-François Alcover, Jul 10 2013

The modern logarithmic integral function li(x) = Integral_{t=0..x} (1/log(t)) replaced the Li(x) = Integral_{t=2..x} (1/log(t)) which was sometimes used because it avoids the singularity at x=1. This constant is the offset between the two functions: li(2) = li(x) - Li(x) = Integral_{t=0..2} (1/log(t)). - Stanislav Sykora, May 09 2015