Le 17-avr.-06, à 19:53, Tom Caylor a écrit :

> A couple of quick thoughts "out loud".
> My previous thought on the possible connection between the OR/AND dual
> (along with addition/multiplication) and the Riemann Hypothesis might
> be extended by looking at the Riemann zeta function.  Notice that the
> infinite series form of the zeta function's uses addition and the
> infinite product uses multiplication.  Could it be that the zeta
> function somehow gives information on observer moments?  (Or instead of
> the zeta function, maybe it would end up being something more general
> function like the Dirichlet-L series  which appears in the generalized
> Riemann Hypothesis.)

This would follow if, as I suspect, the primes encode globally the  
incompleteness mess.
This is possible because the primes gives the whole multiplicative  
structure of all numbers, and the position of the primes (in the  
natural number litany) gives their additive structure, and  
incompleteness can be shown to arise from the mixing of addition and  

> Suppose an observer moment is specified by s, or maybe the imaginary
> part of s. One possibility is something like this, heuristically:  The
> zeta series could describe the 1st person pov, and the zeta product the
> 3rd person pov.
> For the 1st person pov, each term in the series, 1/n^s, could somehow
> correspond to the probability of having a particular "next observer
> moment".  Then the whole series describes the probability of (observer
> moment #1) OR (observer moment #2) OR... (ad infinitum?, with exclusive
> ORs).
> For the 3rd person pov, each term in the product, 1/(1-p(^-s)), could
> somehow correspond to the probabilities of NOT having a particular
> "next observer moment".  Then the whole product describes the
> probability of (NOT observer moment #1) AND (NOT observer moment #2)
> AND... (ad infinitum?, with somehow finally having a selection of ONE
> observer moment out of the infinite(?) possible next observer moments).
>  The equating of the series and product forms is analogous to a
> deMorgan's law (however with an exclusive OR and the ANDing of one
> affirmative selected observer moment).

Remember that the p in the product are the primes. I think you could be  
right in some sense, but hardly in a literal sense.

A link between primes and (many-valued) logic is given by Karpenko:

see intro-pdf:

> A possible variation on this starts with the observation that the zeros
> of the Riemann zeta function lie in 0 < Re(s) < 1 and are symmetrical
> about the critical line, Re(s) = 1/2.  Could it be that the real part
> of s corresponds to the probability of a next observer moment given a
> current observer moment specified by the imaginary part of s?

I do suspect that the real part of the zero corresponds to some ideal  
self-duplication (P = 1/2), but this is not enough clear to me so as to  
put the ideas in words.

>  The
> zeros of the Riemann zeta function could somehow describe the 1st
> person indeterminacy, with the Riemann Hypothesis corresponding to a
> 50/50 chance.

Yes. But how to justify the relation with the 1st person precisely?  
Here I think Voronin's theorem could provide some light by localizing  
form of universal behavior in zeta. (That could lead to a confirmation  
that the global info in zeta does code the incompleteness, but this  
lead to technical difficulties (including finding some old papers!)).
"1/2 + it" is perhaps not the most fundamental vertical line of the  
strip. "1/4 + it" also.

> Along those line, I notice that Chaitin (referencing du Sautoy) says
> that if it could be proved that the Riemann Hypothesis is undecidable
> then it is true, since if it were false then it would be decidable by
> finding a zero off of the critical line.

Yes, by the very work of Riemann, Lehmer, ... Turing, ... Odlyzko. If a  
(non trivial) zero is not on the critical line it must be found by a  
searching program (perhaps in a googol years, or a googol^googol years,  
or later ...).  So, if someone prove that RH (Riemann Hypothesis) is  
undecidable in any very weak theory (far simpler than a lobian theory)  
then RH is true. Indeed.

> (http://maa.org/features/chaitin.html). But could it be that the
> Riemann Hypothesis follows quantum indeterminacy in something like the
> following way?  Just role-playing:  "The Riemann zeta function does
> indeed have zeros which are off of the critical line (or even, it has
> zeros having real parts taking on every real value between 0 and 1.)

1 is already excluded. Actually the width of the critical strip where  
non trivial zeros are yet allowed has been made shorter an shorter.  
Alas, not enough to prove RH.

> This is the non-computable truth.  However, whenever a zero of the
> Riemann zeta function is actually computed (observed), it falls on the
> critical line."

I let you take the full responsibility of this statement :)

Are you introducing a wave collapse in Platonia ???

> Just having fun,

A 18 april joke!  Thanks for the fun and the ideas.

BTW, here is a webpage by Lars Kindermann illustrating that simple  
sequences of numbers can give rise to ... baroque music!  (and others):


Some pieces, mixing such sequences, are quite impressive! Look, I mean  
listen to this:




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