OFFSET
1,3
COMMENTS
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
REFERENCES
S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 425.
A. E. Ingham, The distribution of prime numbers, Cambridge, 1932, p. 3.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Logarithmic Integral
Wikipedia, Logarithmic integral function
EXAMPLE
1.0451637801174927848445888891946131365226155781512015758329...
MATHEMATICA
RealDigits[ LogIntegral[2], 10, 105][[1]] (* Robert G. Wilson v, Oct 08 2004 *)
PROG
(PARI) -real(eint1(-log(2))) \\ Charles R Greathouse IV, May 26 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Frank Ellermann, Mar 13 2002
EXTENSIONS
Replaced several occurrences of "Li" with "li" in order to enforce current conventions. - Stanislav Sykora, May 09 2015
STATUS
approved
