Re: Logrithmic Integral, Pi(x)

["cino hilliard" <hillcino368@hotmail.com>]

GP/PARI CALCULATOR Version 2.2.7 (development CHANGES-1.844)
      i686 running cygwin (ix86 kernel) 32-bit version
     compiled: Oct 28 2003, gcc-3.3.1 (cygming special)
      (readline v4.3 enabled, extended help available)

Hi Karim,
Recently I had a need for the Logrithmic Integral. I could not find it in 
Pari. Here is my rendition.
I am assuming  Euler() is similar to the Euler2() below which has the zeta 
function. It seems to
be working ok at least compared to Maple. Can you implement an Li, Ei 
function?

It is interesting to note that Li2(x) approximates Pi(x) more closely than 
Li(x) in absolute value
when m = log(log(x)). Also A fast Pi(x) would be welcome. My thinking is an 
optional file based
list of Pi(x) generated on request and then read by the interpreter. This 
brings to mind a prior
post I made asking why can't Pari script read directly from a file vs 
loading it into the environment?

\\ From Methematical Handbook for Scientists and Engineers G Korn and T 
Korn,c 1968,
\\Integral Tables page 969, Int(dx/log(x)) = log(log(x) + log(x) + 
log(x)^2/2/2! +
\\                                                                           
                     log(x)^3/3/3! +...
\\ I added Euler() to agree with Mathematica et al.

Li2(n,m) =
       {
       y2 = log(n);
       y  = 1;
       s=log(y2)+ Euler();
       for(x=1,floor(2*log(n)+m),y=y2^x/x/gamma(x+1);
\\     nesting vs exponentiation here would improve speed
       s+=y;
       );
       return(s)
       }

http://numbers.computation.free.fr/Constants/constants.html

\\Fast convergent Euler-Mascheroni constant from above link
Euler2(n) =
           {
           1 - log(3/2) - sum(k=1,n,(zeta(k+k+1)-1)/(k+k+1)/4^k);
           }

Thanks,
Cino

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