zeta()

["Bill Daly" <Bill.Daly@tradition-na.com>]

I think I now understand what is going on in the PARI implementation of  
zeta(), and I have a couple of comments.

1. In the Euler-MacLaurin sum loop in czeta(), the summand is of the form  
gamma(s+2*j-1)/gamma(s) * B(2*j)/((2*j)!*nn^(2*j)). Using the identity  
B(2*j) = -(-1)^j * zeta(2*j) * 2*(2*j)!/(2*Pi)^(2*j), the summand becomes  
gamma(s+2*j-1)/gamma(s) * -(-1)^j * zeta(2*j) * 2/(2*Pi*n)^(2*j). This  
substitution has two potential advantages:

a) If 2*j > bit_accuracy(prec), we may take zeta(2*j) = 1, while for  
smaller values of j, the values of zeta(2*j) can be precomputed and stored  
in a table.

b) The terms (2*j)! cancel out, eliminating the need to call divgsns() in  
the Euler-MacLaurin loop. The new term (2*Pi)^(2*j) can be combined with  
the existing term nn^(2*j) by defining invn2 as 1/(2*Pi*nn)^2 instead of  
1/nn^2.

I don't have a development environment set up for PARI, so I haven't been  
able to test this idea directly. I have tested it successfully in GP, and  
I can supply the code to anyone who may be interested. The same  
substitution might be advantageous in calculating gamma(), lngamma() and  
psi().

2. In the Dirichlet sum loop, the code calls n_s() for each odd n to  
calculate n^-s, and n_s() does a linear search to find the prime power  
factors of n. It seems to me that if memory is not a constraint, one could  
save the cost of the linear searches by calculating the odd entries in  
tab[] by sieving, i.e. something of the form:

  forprime(p=3,nn,
    p_s = p^-s;
    q = p;
    while (q <= nn,
      r = q;
      while (r <= nn,
        tab[r] *= p_s;
        r += q;
      );
      q *= p;
    );
  );

This assumes that the odd entries in tab[] have all been initialized to 1.

Regards, Bill



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