Re: Correctness of derivnum(X=1/2,zeta(X),41)

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

Le Sun, Sep 26, 2021 at 09:55:02AM +0300, Georgi Guninski a écrit :
> On mathoverflow [1] there is conjecture about the nearest integer to
> [zeta^(k)(1-1/B)]=-B^(k+1)*factorial(k)
> 
> Answer of controversial theoretic result claims the explicit
> numerical counterexample k=41,B=2
> 
> I couldn't compute the counterexample on mpmath with high precision
> due to internal error, but pari agrees the counterexample is correct:
> 
> K=41;B=2;T=derivnum(X=1-1/B,zeta(X),K);(round(T)+B^(K+1)*factorial(K))
> %16 = 3.00....
> 
> Is it plausible that the pari computation is correct with high (what?)
> precision?

You can do it simply using lfun (1 is for zeta):
T=lfun(1,1/2,41)
With 1000 decimals, bot methods gives

-147125767958638791291333376948038279859697032057847807999999997.1425315015410577445776251630330179056617261531573654924025775525100945072805912691199162586379502161808694339802832203059822636280018788315201592445383035025391562492954710956453711089080970654460993384070501082533349491308398015047617437292203396182578983940073265620999323391530855977242700565280151725342741544471253612444262064559148072532572931264190227983616422069803048008112393910024020150914419273267214888165745329608985766310517911071345659817098338306617046481981582963248592552790182688947220495180413839704245191279682322130363852065429106570423877376327088055801719693699145116533700311836584861125324839224939287344389080686638823815010727714998552559245450266980666040657930351337459105837934070509632277859527845691379415781276529389013525663704181732726767751137057378836371711081222657155885119269350395343533330951457239091744105849007873169999602658256031832292868573000929627985877563187398741628823784578843151085

So there is little doubt the counterexample is correct.

Cheers,
Bill.