Re: Dirichlet L-Functions

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Mon, Jul 12, 2010 at 02:07:22PM -0500, Ariel Pacetti wrote:
> 
> A stupid question (from my ignorance): I heard a talk recently by
> Guardia-Montes-Nart about a package they implemented in magma to
> compute factorization of ideals and many other number field
> invariants without computing explicitely the ring of integers (which
> I presume Pari does, right?). They posted the algorithm in arxiv
> 
> http://arxiv.org/abs/1005.4596
> 
> where they can work with quite big number fields (I suggested them
> to implement L-series computations, and answered will do), do you
> (experts in Pari) think it is worth it to have something similar
> implemented in GP? (I saw it has nothing to do with the standard way
> to write down ideals nor integers which might be a LOT of work to
> do, and so far I only can think in computing L-functions and ideal
> factorizations as applications).
> 
> I am just sharing my question with you all....

Last time I check, PARI integral basis algorithm (Zassenhaus/Ford/Pauli/Roblot round4) was
faster than Magma/KANT algorithm.
For example:
nfbasis(x^100-x^75+x^50+2^500,1);
take about 3 seconds.

So I would say this is not a priority for us now (we have more pressing issues to adress first
like the handling of fundamental units).

I do not know you would use it to compute L-series:  this only helps for the ramified primes,
and for large discriminants, the expected convergence of the series is too slow to be carried
out on a computer (even assuming the GRH).

Cheers,
Bill.