Bill Daly on Tue, 16 Aug 2005 20:44:38 +0200

 zeta()

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