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

Dear Georgi Guniski,

> On 26 Sep 2021, at 08:55, Georgi Guninski <[email protected]> wrote:
> 
> 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, 

that is wierd; using python as an interface to mpmath it gives 
languasc@languasc-macmini Desktop % python3 conj.py 70
------ 
precision  70
value =  3.0
languasc@languasc-macmini Desktop % python3 conj.py 80
------ 
precision  80
value =  3.0

The point is that these quantities are large (their order
of magnitude is about 10^62) but their sum is tiny.
So you need to choose a sufficiently large precision 
to perform the computation.

The python-mpmath script I wrote is at the bottom of this file.

An alternative code for gp is the following

? \p100
   realprecision = 115 significant digits (100 digits displayed)
? K=41
%2 = 41
? B=2
%3 = 2 
? round(zetahurwitz(1-1/B,1, K))+B^(K+1)*factorial(K)
%5 = 3.000000000000000000000000000000000000000000000000000000000

Best regards
Alessandro Languasco

> 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?
> 
> What other CAS say about it?
> 
> [1]: https://mathoverflow.net/questions/404779/on-the-nearest-integer-to-zetak1-1-b-b-ge-2

--------------------------------------------------------------
import sys
defaultprecision = sys.argv[1]
import mpmath as mp

# function z 
def z(s,a,k): 
    y = mp.zeta(s,a,k)
    return y

#  print("Give me the desired accuracy (# of decimal digits): ")
#  defaultprecision = input();
defaultprecision = int(defaultprecision);
#print(defaultprecision);
mp.mp.dps = defaultprecision;
mp.mp.pretty = True

print("------ "); 
print("precision ", defaultprecision);
B=2
K=41
print("value = ", mp.nint(z(1-1/B,1,K))+ mp.power(B,K+1)*mp.fac(K));