Re: prodeulerrat

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

Dear John,

* John Cremona [2020-04-08 10:14]:
> I'm using the prodeulerrat() function to compute a value which will be
> published, so I want to give proper credit.  Of course I will credit
> pari/gp itself using the guidelines at
> http://pari.math.u-bordeaux.fr/faq.html#quote (and by the way, I think
> it would be a good idea to have a direct link to that very clearly
> visible on the home page).

Good idea. I'll do that.
 
> I cannot see any specific credit for that function either in the users
> manual or in the code, neither a person's name or a reference to any
> literature.  I know that in Henri's Number Theory vol II there's a
> method described on pages 208-210 (I may be working at home but I do
> have a few essential books with me) and that refers to his
> unpublished/unfinished notes hardylw.dvi, which I also have.

Corolary 10.3.22 in the former is the basic algorithm, yes; PARI/GP
implementation by Henri and me. The method is folklore but I learnt it from
Henri's notes (which were already around in 1996). Beware that the method can
be made rigorous, but our simple-minded implementation is *not* rigorous,
although it will be correct for "most" examples.

To compute \prod_p B(1/p) with proven bounds, we have rigorous truncation
error bounds, but our implementation doesn't bother with round-off errors:
interval arithmetic would be needed there, in particular when computing
B(1.0/p) for a number of "small" primes and when expanding the power series
of log B [in floating point arithmetic...]

If the degree or coefficients of B get huge, it is easy to obtain examples
where none of the given digits is correct.

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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