Le 18-avr.-06, à 18:48, Tom Caylor a écrit :


> I meant to exclude 0 and 1.  I am not aware that there have been any
> subsets of the critical strip 0 < Re(s) < 1 that have been found to be
> zero-free, yet.  I know that it has been proved that at least 40% of
> the zeros are on the critical line.


I am pretty sure the critical strip has been squeezed but I have no 
references under the hand. I will look forward for them and I will let 
you know.




> No.  In the sense that I'm NOT saying:  Given a (fixed) value of y,
> there might be a zero at s=x+iy off of the critical line... until you
> actually compute it at which time you find that it is on the critical
> line.
>
> But, yes, in a different sense, and here I do take full responsibility,
> with tongue in cheek, but not too far lest I bite it.  Perhaps
> mathematicians have been committing an error in the realm of infinity,
> perhaps associated with the Axiom of Choice.


I don't see where. Actually if someone prove RH with any reasonably 
consistent strong axiom, he/she will prove that RH is not refutable by 
Peano Arithmetic (PA), and that would show that RH is true, given that 
if RH is false, PA can prove it with finitary means.
(RH is for Riemann hypothesis)




>  Perhaps there are zeros
> off of the critical line, but they all lie beyond any chosen (observed)
> finite number.  Perhaps the off-critical-line zeros are non-computably
> existent but affect us in tangible ways.  Perhaps the Riemann
> Hypothesis is even *outside* of the realm of undecidable.


I don't know what you mean by *outside* of the realm of undecidable. 
(Note that "undecidability" is a relative notion, unlike 
uncomputability which is absolute (through Church's thesis). I will 
come back on this in my post for George (in may).


Bruno


http://iridia.ulb.ac.be/~marchal/


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