Karim Belabas on Thu, 27 Apr 2023 02:28:54 +0200


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Re: Recognizing numbers using PARI/GP 

* Bill Allombert [2023-04-26 23:51]:
> On Wed, Apr 26, 2023 at 11:01:16PM +0200, kevin lucas wrote:
> > I recently had cause to run a computation that spat out the number
> > 2.0298832… A little research suggested that this should be a special value
> > of a Dedekind zeta function, but I can’t find the exact relation. Now, I’m
> > aware that PARI/GP can recognize algebraic numbers using algdep, and one
> > can easily incorporate numbers involving common constants like $\pi$ and e
> > with qflll. But what if you suspect relations between special values of
> > special functions (e.g. eta/gamma/zeta functions) and you don’t know which
> > values of which functions, as in this case? I’ve known mathematicians who
> > found relations like this all the time, which leads me to believe there are
> > some dark arts in PARI for this that are only well known within a small
> > community.
> 
> Dedekind zeta function special value are multiplicative in nature, so you
> might have more chance by taking the logarithm and use lindep with the
> logarithms of special value of L functions.
> 
> If your number if given by a series, the shape of the series gives a tip.
> For example, if your series is sum a_n/n^2 with integral algebraic integers a_n,
> then you should use lindep with:
> values at 2 of L function, dilogarithms, and product of value at 1 of two L
> functions or logarithms.
> 
> You give too few decimals to try this.
> 
> However, Google suggests <https://en.wikipedia.org/wiki/Hyperbolic_volume>
> which gives:
> 
> ? -6*intnum(x=0,Pi/3,log(2*sin(x)))
> %53 = 2.0298832128193072500424051085490405719
> 
> Plouffe ISC suggests:
> 
> ? hypergeom([1,1,1]/2,[3,3]/2,1/4)*2
> %54 = 2.0298832128193072500424051085490405719

>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

Cheers,

    K.B.
-- 
Pr Karim Belabas, U. Bordeaux, Vice-président en charge du Numérique
Institut de Mathématiques de Bordeaux UMR 5251 - (+33) 05 40 00 29 77
http://www.math.u-bordeaux.fr/~kbelabas/