@janhardo 
Can I also do that check on the cauchy riemann equations via : evalb ?

if evalb(u_x = v_y) and evalb(u_y = -v_x) then
    printf("The function is analytic (holomorphic) at this point.\n");
    printf("The derivative f'(z) is %a + I*%a\n", u_x, v_y);
else
    printf("The function does not satisfy the Cauchy-Riemann equations and is not analytic.\n");
end if

@Carl Love 
Thanks, Yes, that "is" command is essential to establish that the cauchy-riemannt equations are true for all variables

@Axel Vogt 
Thanks, forms a nice basis to rewrite the procedure and the hereby use the D operator instead of diff command.

@acer 
Yes, when to use a branch from a topic?
There are several ways to derive that zeta(2) value. 
I thought this was 1 branch to explore this further and another branch for a different approach to derive zeta( 2).
Otherwise it will be one long thread of proof methods topics , fine by me too and forget the branches.

How about checking this with ODESteps in Student package ?

@vv 

Thanks, I think you are absolutely right to study the methods of evidence.
The difficulty for me is still how to start translating this into Maple, and your elaboration is an example of how it can be done.

Don't make any progress yet , let me ask Maple AI to come up with the code variants to solve the Basel problem ( The Zeta (2) value ). 
Note: get the impression that the standard AI 3.5 works better for Mathematica ? (not the plugin yet ) 
May try this for the Basel problem for Maple as well...

@janhardo 

It is about comparing series expansion and product expansion of Sin[x]/x. The coefficient of x^2 in the series must coincide with coefficient of x^2 in the product (after expanding the product and collecting powers of x).
Seems to me one of the proofmethods...

simply this equation?
 

eq1 := Product(1 - x^2/(n^2*Pi^2), n = 1 .. infinity) = Sum((-1)^n*x^(2*n)/(2*n+1)!, n = 0 .. infinity) = sin(x)/x;

@sand15 
 

"More generally, ChatGPT does not shine for its clarity (Point 6 is quite obscure)."

You can challenge chatGPT to come up with a better answer and also zoom in on a detail 
I'm using the public chatGPT 3.5 (free) and not the one in Maple itself and therein is a better version chatGPT 4.5?, but don't find it working conveniently in Maple yet 
Also mathematica has developed its own language model plugin and that requires chatgtp 4.5 with a subscription
I wonder how this works in Mathematica 
Ideal would be if Maple would use its own language model focused on Maple content and then you can ask anything, at least it seems to me that this is possible ?

@Axel Vogt 
Thanks, those are great sources of information on the Basel problem.
Don't know which double integral of Apostle that is ?
Via the Student package in Maple with a step solution for 1 integral ( surely a double integral can be solved with repeated integration ? )

@mmcdara 
Thanks, there are not that many faculties numbers to investigate for a given number whether it is a faculties number 

Oh now I see you replied that in your opinion, no faculties numbers can be written symbolically in the series
Well, yes , I do recognise some of them now, so for a proof step well follow then? 

@sand15 
Thanks!, you are quite right that chatGPT is not a really solid foundation of knowledge, but I still find it very helpful to figure out some.
From orginele Euler pdf is quite a chore to get exactly clear why the sum: 1/n^2 (to infinity) = 1/6 Pi^2
Still try in Maple with the wiki examples

Let's look at sin(x) 
Now how to get n! numbers in the serie?

T:=(x,n)->subs(t=x,convert(taylor(sin(t),t=0,n+1),polynom));
   T := proc (x, n) options operator, arrow; subs(t = x, 
      convert(taylor(sin(t), t = 0, n+1), polynom)) end proc


sin(x)=T(x,10)+`...`;
                  1  3    1   5    1    7     1     9      
     sin(x) = x - - x  + --- x  - ---- x  + ------ x  + ...
                  6      120      5040      362880         

@vv 
Thanks, 
"but note that Maple cannot use ithprime symbolically."

I don't think it's a problem, because from comparing 2 finite sequences, Euler could derive something  
So first get the two series according to me.
From the euler product formula you have given, a partial sum can be made 

"1. primes (wrongly suposed to be the set of prime numbers in Maple) is infinite, so it cannot work for Product. Maple rejects even the syntax.
This is euler product formule ....
From this defintion of the zeta function as an euler -produkt formula, this maple code cannot make out the series... should be possible I suppose
Possibly the hypergeometric function is related to the eulerprodukt formula ?