1,1

In computing Li(x) we can limit the iterations to 2*log(x) + m where m is suitably large to allow convergence to the precision desired. If we let m = floor(log(log(x))) we get a better approximation of Pi(x) than the full Li(x) expansion. With this m we get Li(x) < Pi(x) often but still closer in absolute value to Pi(x). Note the use of the gamma function to quickly compute factorials in the precision range i.e. gamma(x+1) = x!.

Xavier Gourdon, Collection of approximations for pi

The logarithmic Integral can be computed by Li(x) = log(log(x)) + log(x) + log(x)^2/2/2! + log(x)^3/3/3! + ... + 1 - log(3/2) - sum(k=1, prec, (zeta(2k+1)-1)/(2k+1)/4^k). This last expression is a fast converging series taken from the link for the Euler-Mascharoni constant 0.57721.. where prec is the precision level you are using. PARI has an Euler() function built in so that was used in this calculation.

(PARI) pw2pix(n, m) = { for(x=1, n, y=10^x; print1(floor(Li(y, m))", ") ) } Li(n, m) = { y2 = log(n); y = 1; z=1; s=log(y2)+ Euler(); for(x=1, floor(2*log(n)+m), y=y2^x/x/gamma(x+1); s+=y; ); return(s) }

Sequence in context: A135490 A110706 A001341 this_sequence A057754 A001473 A063888

Adjacent sequences: A089893 A089894 A089895 this_sequence A089897 A089898 A089899

nonn

Cino Hilliard (hillcino368(AT)gmail.com), Jan 10 2004