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Bill Allombert on Thu, 27 Apr 2023 11:52:53 +0200
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Re: Recognizing numbers using PARI/GP
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: Recognizing numbers using PARI/GP
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Thu, 27 Apr 2023 11:51:38 +0200
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On Thu, Apr 27, 2023 at 02:27:13AM +0200, Karim Belabas wrote:
> From Bill's first formula (and Milnor's proof of it given in the
> Wikipedia article), you can express this in terms of Lobachevsky's function
> and in turn get your expected relation to Dedekind zeta function:
>
> ? lfun(x^2+3,2)/zeta(2) * sqrt(27) / 2
> %1 = 2.0298832128193072500424051085490405719
So you see it is a multiplicative formula as expected, so you could find it with
? lindep([log(z),log(lfun(-3,2)),log(zeta(2)),log(2),log(3)])
%71 = [-2,2,0,-2,3]~
so z^2 = lfun(-3,2)^2*2^-2*3^3 and
z = 3*sqrt(3)/2*lfun(-3,2)
Cheers,
--
Bill Allombert
Ingénieur de recherche en calcul scientifique ❄
CNRS/IMB UMR 5251