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Programming details on the computation of the Riemann zeta function using Aribas

  1. Oct 2, 2011 #1
    (1) Let s be a complex number like s = a + b i, then we define [tex] \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} [/tex]

    Our aim:

    to compute ζ([itex]\frac{1}{2}[/itex]+14.1347 i) with the help of the programming language Aribas

    (2) Web Links

    Aribas: http://www.mathematik.uni-muenchen.de/~forster/sw/aribas.html

    Dirichlet eta: http://en.wikipedia.org/wiki/Dirichlet_eta_function

    Euler transformation of alterning series: http://en.wikipedia.org/wiki/Euler_transform

    (3) Books: Knopp: Theorie und Anwendung der unendlichen Reihen; Henrici: Applied
    and Computational Complex Analsis; Derbyshire: Prime Obsession

    (4) Basics: The Riemann zeta in (1) is defined only for s.Re > 1 (i.e. the real part of s);
    we use the 'alternating zeta' or Dirichlet eta defined as

    [tex] \eta(s) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s} [/tex]

    defined for s.Re > 0

    The Riemann zeta and the Dirichlet eta are related through:

    ζ(s) := [itex]η(s) / (1-2^{1-s}[/itex])

    With the Euler transformation of the Dirichlet eta, the Riemann zeta looks like:

    [tex]\zeta(s)=\frac{1}{1-2^{1-s}}
    \sum_{n=0}^\infty \frac {1}{2^{n+1}}
    \sum_{k=0}^n (-1)^k {n \choose k} (k+1)^{-s} [/tex]


    To be continued with the programming details
     
  2. jcsd
  3. Oct 2, 2011 #2
    Aribas is a computer program for number theoretical calculations.
    It comes with:
    - a built-in large integer arithmetic
    - several modes for floating-point accuray; single/double/long (32/64/128 bit) and user-defined bit number
    - a bunch of number theoretical functions
    - a great variety of example program scripts
    - the possibility to easy implement additional user-written functions

    For the Riemann zeta calculations, we have to implement a complex arithmetic,
    which is easily done by a handfull of functions; we need especially;

    AddC(a,b) to add two complex variables, SubC(a,b), AbsC(a), which
    can be programmed at a very basic programming level.

    A little bit more is needed for a function to calculate the
    exponentiation of a real base raised to a complex exponent.

    As an example, here is the Aribas code for 'PowRC':
    (a complex variable c is stored in an array[2] of reals,
    the Re part in c[0], the Im part in c[1])

    Code (Text):

    function PowRC(b: real; a:complex): complex;
    var
       c: complex;
       x,y: real;
    begin
       x:=log(b);
       y:=x*a[1];
       x:=exp(a[0]*x);
       c[0]:=x*cos(y);
       c[1]:=x*sin(y);
       return(c);
    end.
     
    To be continued with the further programming details
     
  4. Oct 3, 2011 #3
    Code (Text):
    Deta(x,n)={
        my(b=2^(2*n-1), c=b, s=0);
        forstep(k=n-1,0,-1,
            s += c*(-1)^(k)/((k+1)^x);
            b *= ((2*k+1)*(k+1)) / (2*(n+k)*(n-k));
            c += b;
        );
        return(s/c);
    }
    Rzeta(s,n=100)=Deta(s,n)/(1-2^(1-s))
     
  5. Oct 3, 2011 #4
    Aribas code for the Dirichlet eta function:

    Code (Text):

       i:=1; t:=(0.0,0.0);
       for n:=0 to 9999 do
          i:=i*2;      
          x:=(0.0,0.0);
          for k:=0 to n do
             y:=PowRC(1.0/(1.0+k),s);
             j:=nCr(n,k);
             y:=j*y;
             if odd(k) then
                x:=SubC(x,y);
             else
                x:=AddC(x,y);
             end;
          end;
          x:=x/i;
          t:=AddC(t,x);
          if AbsC(x) < eps then
             [B]break[/B];
          end;
       end;
       return(t);
     

    The for loop will terminate after 9999 iterrations, but in practice, the break
    will be effective miuch earlier

    [tex]{n \choose r}[/tex] the binomial coefficient is computed with the function 'nCr' ("from n choose r")

    Numerical results for s = [itex]\frac{1}{2}[/itex]+14.1347 I

    eps = 1.00000000000000000e-13

    eta2: -0.000002874863 - 0.000047246855 I

    factor: 0.411257843979 + 0.091385435419 I

    RiemannZeta(s): 0.000003135364 - 0.000019693360 I
     
  6. Oct 3, 2011 #5
    realprecision = 28 significant digits
    eps=1e-13
    zeta embeded function in PARI/GP
    Code (Text):
    s=1/2.+I*14.1347
    abs(YourZeta(s)-zeta(s))==2.4718664606232304313 E-14  60 iterations 219 mSec
    abs(MyZeta(s)  -zeta(s))==1.6630931417336143861 E-15  30 iterations  15 mSec  
    abs(zeta(s)    -zeta(s))==0.E-33                      ??              0 mSec
     
    Last edited: Oct 3, 2011
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