Re: Recognizing numbers using PARI/GP

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

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