Re: Recognizing numbers using PARI/GP

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

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

Cheers,
Bill.