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Euler-Maclaurin formula for zeta in PARI/gp

for(X=1,khyber,t=t+bernfrac(2*X)*q;q = q*((s+2*X-1)*(s+2*X)/(N*N))/((2*X+1)*(2*X+2)) );

t = t + exp((1-s)*log(N))/(s-1) + exp(-s*log(N))/2 ;

g = sum(X=1,N-1,exp(-s*log(X))); z2 = t+g; abs(z2)

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Written by meditationatae

May 4, 2013 at 9:14 pm

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  1. Preceded by:

    ? N=40000
    %11 = 40000

    ? khyber=40000
    %12 = 40000

    ? t=0;

    ? q=s/(2*N*exp(s*log(N)));

    and
    using:
    ? \p
    realprecision = 80011 significant digits (80000 digits displayed)
    and a 70,000-digit approximation to rho_1 = 1/2 +I*14.13….

    –> Newton’s method to locate zero.
    –> derivative approximated by difference quotient

    memo for self:

    [david@localhost pi_Math]$ pwd
    /home/david/pi_Math

    [david@localhost pi_Math]$ ls -l
    total 1988
    -rwxrwxr-x. 1 david david 8539 Dec 18 00:01 a.out
    drwxrwxr-x. 2 david david 4096 Mar 30 05:12 MD5
    -rw-rw-r–. 1 david david 225 Feb 26 21:12 mygpzetaformula.txt
    -rw-rw-r–. 1 david david 154131 Dec 18 03:23 my_pi_test01.txt
    -rw-rw-r–. 1 david david 162907 Feb 5 15:59 PARIrefcard.pdf
    -rw-rw-r–. 1 david david 1643 Dec 17 23:57 pi001a.c
    drwxrwxr-x. 2 david david 4096 Feb 9 15:56 poisson
    drwxrwxr-x. 4 david david 4096 Mar 16 13:11 rho
    -rw-rw-r–. 1 david david 1577359 Feb 5 16:15 users.pdf
    -rw-rw-r–. 1 david david 70010 Feb 25 21:21 zeta70K
    drwxrwxr-x. 2 david david 4096 Dec 26 10:56 zeta_Math
    drwxrwxr-x. 3 david david 4096 Dec 27 01:49 zeta_math2
    drwxrwxr-x. 3 david david 4096 Dec 31 04:43 zeta_math3
    drwxrwxr-x. 3 david david 4096 Dec 31 22:36 zeta_math4
    drwxrwxr-x. 3 david david 4096 Jan 7 02:02 zeta_math5
    drwxrwxr-x. 2 david david 4096 Jan 18 01:26 zeta_math6
    drwxrwxr-x. 2 david david 4096 Jan 20 16:53 zeta_math7

    meditationatae

    May 4, 2013 at 9:28 pm


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