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Euler-Maclaurin summation for high-precision zeta computation

I wrote about this yesterday in more detail in sci.math .

Using 1st formula of Section 6.4 in H. M. Edwards’ book “Riemann’s Zeta Function”.

Attempting to find near-optimal parameters N and ‘nu’ so
as to minize time to compute zeta(1/2 + i*14.13…).

This case: 60,000 decimal digits precision after “0.” required.

Tested:

N = 30,000 and ‘nu’ = 35000.
Time spent on computation: 15 hours 17 minutes,
line before output %671 below.

? write(rho1b60k, s802);

? s=s802;

? t=0; q=s/(2*N*exp(s*log(N))); 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)
time = 15h, 16min, 39,337 ms.
%671 = 5.131221963508608095 E-60010

? N
%672 = 30000

? khyber
%673 = 35000

 
$ date
Sat Feb 16 12:59:32 EST 2013

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Written by meditationatae

February 16, 2013 at 6:09 pm

Posted in History

Tagged with ,

2 Responses

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  1. We have a lower time of computation of 14 hours 26 min
    when, in the notation of H.M. Edwards’ in his book,
    N = 32,000 and ‘nu’ = 32,405 :

    ? N
    %682 = 32000

    ? khyber
    %683 = 32405

    ? t=0; q=s/(2*N*exp(s*log(N))); 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)

    %684 = 2.777212228894781921 E-60007

    time = 14h, 25min, 44,279 ms.

    ————————————
    Mon Feb 18 06:38:42 EST 2013

    meditationatae

    February 18, 2013 at 11:39 am

  2. The choice N = 33000 produces a marginally better time:

    ? N
    %11 = 33000

    ? khyber
    %12 = 31560

    ? t=0; q=s/(2*N*exp(s*log(N)));

    ? 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)

    time = 14h, 15min, 18,468 ms.
    %14 = 9.774170032632170803 E-60000

    $ date
    Thu Feb 21 17:04:09 EST 2013

    meditationatae

    February 21, 2013 at 10:04 pm


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