Re: Number Theory - Study methods
- From: Angus Rodgers <twirlip@xxxxxxxxxxx>
- Date: Sun, 28 Oct 2007 23:48:34 +0000
(Apology: I was rather reluctant to go into this at all -
after the drubbing I received last time! - and then the
message became worryingly long, in spite of my efforts to
keep it brief, and I'm afraid it now reads poorly because
some parts near the end were composed before some parts
near the beginning. Oh well, it's only Usenet, and at
least I'm not top-posting.) :-)
On Sun, 28 Oct 2007 20:15:20 -0000, Jeremy Boden
<jeremy@xxxxxxxxxxxxxxx> wrote:
BTW I don't understand your point (viii) - Apostol's use of
O(g(x)) seems OK -
It's quite standard to quantify out the number a. The trouble is
that some of Apostol's arguments require a to equal 1 ("locally",
so to speak). Also, some authors do take the trouble to specify
the value of a. Some people in the discussion took the position
that in fact Apostol /was/ specifying the value of a, but he just
hasn't made this completely clear in the wording of his definition
of O(). Other good points were made against what I was arguing
(including one by David Ullrich). The whole thing wasn't clear-cut,
and it would be quite hard work to dig it all up again and write
out a fair summary. There were perhaps four or five different
positions taken, all of them with something to be said in their
favour (here excluding one or two that were incoherent). I was
very surprised to encounter any argument at all! It had seemed
a very simple point that I was making, just pointing out a minor
flaw in an excellent textbook. It's surprising (to me, at least)
that even mathematics can be controversial (and even heatedly so).
Now, Lemma 7.5 is actually the most complex case of all. (Do you
really want to go into this?!) The other problems I identified
were all, I think, quite "local": what the reader had to do was
to realise that in certain proofs one had to use an O() estimate
where the value of a was fixed at 1, and then one had to go back
to check that Apostol had actually justified such an estimate -
which in every case he had, so the problem was solved. It was
a somewhat annoying chore to have to do this, but basically one
only had to connect two theorems each time (the one where the O()
estimate was obtained, and the one where it was subsequently
used).
If I'm not mistaken about Lemma 7.5, fixing this error requires
a minor modification to Apostol's proof of Dirichlet's theorem
at several points. Conceptually, nothing is wrong at all: one
just has to adjust a few formulae, and the proof becomes valid.
So one can easily ignore the problem (or not notice it in the
first place), and nothing of any real importance is lost.
I don't know if I'm mistaken, because I got no replies to my
message in the OU M823 conference (or the e-mail to my tutor)
identifying the problem. So I'll be very brief (and then we
can discuss it at length if you want to).
and existentiality would appear to be satisfied since there would be
always seem to be a least value for "a",
(Not always, and not in the present case, because there is
of course no least value of a such that a > 1.)
OK, I've looked up my notes on Lemma 7.5. (Bear in mind, though,
that the difficulty here is not the only one. It may even have
been someone else who first pointed this one out, in response to
what I had written - which was mostly or entirely about the case
where one has to take a = 1 for the argument to be valid.)
(Checking back through the correspondence, I see that: I first
encountered a problem in Chapter 3, I posted about it, a long
and quite heated discussion ensued, Paul Pollack posted on 9 Feb
2005 agreeing with my point, and identifying a further similar
difficulty in Chapter 6, which I hadn't reached yet; and when
I got to Chapter 7, I encountered a difficulty which could be
traced back to the problem that Dr. Pollack had identified.
But by this time, I was no longer in any discussion about the
issue, and my attempt to raise it with my fellow-students and
tutor fell flat; hence my own thoughts on the matter have had
no "peer review".)
Shall I go into this now? It's really quite technical (never
mind the fact that I haven't studied analytic number theory
in the last 2 years, and I have a head like a sieve - no pun
intended!), and when you refer below to "the lemma mentioned",
I'm not entirely sure that you mean Lemma 7.5 (although you
probably do).
(If you can cope with this, you'll have no trouble with the
rest of M823! - The converse is not true, don't worry about
that!)
VERY briefly, then:
The crux is equation (13) on page 151. Here we have x >= 1,
and c and d are integers such that cd <= x.
On the grounds that x/d >= 1, we are asked, in the next line
of the proof, to infer that a certain sum for c <= x/d may be
replaced by:
-L'(1,xi) + O(log(x/d)/(x/d))
The justification is the estimate contained in formula (10)
of Theorem 6.18 (which is the one identified as problematic
by Paul Pollack): the "x" in Theorem 6.18 (10) is simply to
be replaced by the "x/d" in Lemma 7.5 (which is >= 1).
Apostol states Theorem 6.18 for all x >= 1. But formula (10)
of this theorem (the other two, (9) and (11), are OK) is only
valid for all x >= a, where a > 1. The exact value of a is
immaterial, apart from this constraint.
The problem with (10) is that, for 1 <= x < 2, the LHS is zero,
but the first term on the RHS is non-zero (it is a constant,
independent of x), therefore the difference between the LHS and
the constant on the RHS cannot be bounded by a constant multiple
of log(x)/x for all x >= 1, because of course log(1)/1 = 0.
That's the /cause/ of the problem with Lemma 7.5. So far,
this is just a "local" problem, like all the others (which
I haven't bothered to discuss here). But the /effects/ of
the problem extend throughout Apostol's proof of Dirichlet's
theorem.
Let me just reiterate at once that the "fix", once the problem
has been identified, is minor and easy. (Perhaps also a matter
of taste - I had my own way of fixing it, but no-one discussed
the matter with me, so I don't know of any other fixes which
might have been more aesthetic.) And let me also reiterate
that it's quite possible that I'm mistaken about Lemma 7.5
(although surely not about Theorem 6.18), because I haven't
been forced by argumentative people to reexamine my argument.
I don't think there's any need to go into the repair of the
proof of Dirichlet's theorem here (but I'll do so if anyone
is interested).
It remains only to add (in case you're worried) that the
/conclusion/ of Lemma 7.5 is still true without modification
(but this is not obvious - at least it did not seem obvious
to me when I was last thinking about it).
although that proof of the lemma
mentioned does look a bit rushed. However, I appreciate that David
Ullrich can be quite tenacious - so I'll wait and see..
You're in good company. Not only David Ullrich (who really
knows his stuff, by the way) but almost everybody with whom
I discussed this point disagreed with me. However, one does
not reach a rational judgement by counting heads - however
wise! No doubt you will make up your own mind about it, when
you come to the relevant part of the course (or may not even
be troubled by what troubled me, which is fine, because no-one
else on the course in my year was troubled by it either). In
my defence (but without rehashing the actual arguments, which
would be pointless) I'll just say that the author of another
book on analytic number theory posted to say that I was right,
and that he'd been held up by exactly the same point when he
was reading Apostol! So even if he and I are both wrong, I'm
also in good company. :-)
--
Angus Rodgers
(twirlip@ eats spam; reply to angusrod@)
Contains mild peril
.
- Follow-Ups:
- Re: Number Theory - Study methods
- From: Jeremy Boden
- Re: Number Theory - Study methods
- References:
- Number Theory - Study methods
- From: Jeremy Boden
- Re: Number Theory - Study methods
- From: Angus Rodgers
- Re: Number Theory - Study methods
- From: Jeremy Boden
- Number Theory - Study methods
- Prev by Date: Re: Difference between partial derivates?
- Next by Date: Re: Series Expansion of 1/x ?
- Previous by thread: Re: Number Theory - Study methods
- Next by thread: Re: Number Theory - Study methods
- Index(es):
Relevant Pages
|
