Re: Factoring and rationals

From: Robert Maas, see http://tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 04/17/05 

Date: Sun, 17 Apr 2005 11:53:59 -0700

> From: "Tim Peters" <tim.one@comcast.net>
> do you think Dirichlet was deluded in some way when he proved that
> 6/pi^2 is the probability that two integers chosen at random are
> coprime?

No, I think if he did what you say (I am not going to spend the next
half hour doing a Google search to verify that you got the facts
correct, I'll just trust you for sake of argument) he carefully defined
what he meant by that probability, and then proved that such
probability defined in that way was indeed 6/pi^2.

> In number theory, a claim that this-or-that probability holds for an
> integer "chosen at random" has a conventional, rigorous meaning: you
> consider the claim applied to an appropriate finite set of integers
> (1 through N is most common), and compute the probability wrt the
> uniform distribution on that finite set. Then you look at what
> happens as N (the size of appropriate sets) goes to infinity. If a
> limit (in the ordinary sense of the word) exists, then "the
> probability for an integer chosen at random" is, by agreement, that
> limit.

That's no good, because that method covers only the positive integers.
By that method you could conclude that 100% of the integers are greater
than zero. So in this case I don't trust what you wrote because it's
clearly wrong. Try correcting your mistake and I'll consider your
revised definition.

Note that in general, you define an arbitrary enumeration of the
non-finite set of interest, which works only if the set is countable in
the first place, then you apply the limit process you outlined above.
(Of course you must have an enumeration of the *whole* set, not just
part of it as you mistakenly did.) But note that this method is
specific to a given specified enumeration, and wouldn't apply to other
enumerations that are possible or even plausable or even rational or
even equally reasonable as the given enumeration. In the case of
positive integers, the interval [1..n] where n gets larger seems
reasonable as a "canonical" enumeration, but it's not the only
enumeration, not by a long shot. If you wanted to enumerate all the
integers, not just the positive integers, there are several equally
plausable enumerations to choose from. For example, start with 0, then
alternate 1 -1 2 -2 etc. Or go the other way 0 -1 1 -2 2 etc. Or go
back and forth symmetrically: 0 -1 1 2 -2 -3 3 4 -4 etc. Or do them in
blocks by number of bits in the magnitude, alternating entire blocks: 0
1 -1 2 3 -2 -3 4 5 6 7 -4 -5 -6 -7 etc. Except for that last one, I
believe all of them would produce the same result whenever any one of
them converges.

When you try to enumerate the rationals, however, there's no particular
method that is rightly canonical, i.e. no way to choose among the
various likely candidates, and I don't believe the equally-good choices
give the same limit even where they all give some limit, but I'm not
sure about that. Note that the Subject of this thread is something
about rationals, not integers, not positive integers.
Note from the original poster:
> There are an infinite number of factorizations into two factors that
> are rational for j^2 T.
I.e. he's talking about an infinite set of rationals.
> Now considering those infinities, you can break them up into what we
> call trivial and non-trivial factorizations easily enough,
I.e. he's splitting a set of rationals into two subsets.
> One poster made a telling reply where he claimed that there were more
> trivial factorizations than non-trivial ones.
See, there's the problem: What particular enumeration of this infinite
set of rationals is canonical or just taken for granted when estimating
which fraction of the totality is in one subset or the other? If one
set is finite then it's clear, 100% are in the infinite set and 0% are
in the finite set. But if both are countably infinite, what??

Relevant Pages