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Experimenting with Euler-McLaurin summation for zeta function

testing 123..

meditationatae

If we look at MathWorld at the page for the Bernoulli numbers, we find one analytic formula for the even Bernoulli numbers (eqn. 41 or 42).

I’ve been using this analytic expression in formula 1 of Section 6.4 of Edwards’ book on the Riemann zeta function.  I can do this with PARI/gp.  One term that requires a lot of time to

compute is sum_{n = 1, … N-1} n^(-s)  ;  the analytic formula for the Bernoulli numbers doesn’t help with that “partial sum” term.  Where the analytic formula will help is in the terms involving the Bernoulli numbers.  For increased precision, one needs both more terms in the “partial sum” and more Bernoulli-number terms. So I’m experimenting. “khyber” stands for Edwards’ “nu”, roughly the number of even Bernoulli numbers included.

 

? khyber

%185 = 37000
? N=N+1000

%186 = 16300   // value of N

? t=0;
  q=s/(2*N*exp(s*log(N)));
 …

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Written by meditationatae

May 29, 2014 at 7:46 am

Posted in History

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