Re: Problem with `big oh' estimates in number theory

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/10/05 

Date: Thu, 10 Feb 2005 07:24:55 -0600

On Wed, 09 Feb 2005 16:27:16 +0000, Angus Rodgers
<angus_prune@bigfoot.com> wrote:

>On Wed, 09 Feb 2005 08:39:08 -0600, David C. Ullrich
><ullrich@math.okstate.edu> wrote:
>
>>On Wed, 09 Feb 2005 13:16:00 +0000, Angus Rodgers
>><angus_prune@bigfoot.com> wrote:
>>
>>what seems most likely
>>to me is that you're overlooking facts that he's using
>>without stating explicitly because he thought they
>>were obvious.
>
>No, he's using the fact that certain terms, which are
>functions of the number q = x/d, are bounded for *all*
>values of q >= 1, and not just for all `sufficiently
>large' values of q. The lower bound for `sufficiently
>large' *has* to be 1, for these proofs to work at all.
>
>My point is that this fact is essential to all three
>of his proofs, but it has never been stated by him.

I followed all of that except for the word "no". You're
_saying_ that he's using certain facts that are not
explicitly stated.

>This isn't just some `obvious' point, which he expects
>the mathematically mature reader to work out for him/
>herself.

But _you_ were in fact _able_ to fill in the gaps,
even though they can't be something that he's
expecting the reader to be able to do.

Wow.

>[...]
>
>It is indeed bounded, but it is not *clearly* bounded,
>and that is my whole point. The proof that the terms
>in question (in the three proofs) are bounded for all
>values of the argument >= 1 is *exactly* the same as
>the proof given by Apostol that they are O(foobar).

Let's stop and think about this for a second. We have
a crucial missing fact. And you just _said_ that
an argument given in the book is sufficient to
establish that crucial missing fact. But in spite
of the fact that the missing fact follows from
explicitly stated arguments the truth of the
missing fact doesn't count as obvious?

Never mind, there's a more relevant question below:

>[...]
>
>>For example you say he gives preliminary results on
>>zeta(s), of the form
>>
>> zeta(s) = main terms + O(error terms).
>>
>>Do the main terms there blow up like 1/(s-1) as s decreases to 1?
>
>No,

Really? Exactly what is the statement he makes about zeta(s)
that you're referring to here?

A possibility occurs to me. Your comments below make more
sense if I assume that you misread my question, as though
I was asking whether the error terms in the stated estimate
blew up. The reason it looks that way is that error terms
blowing up would be bad, while the main terms blowing
up, _exactly_ like 1/(s-1), would be good - you're saying
"no it doesn't blow up" as though that were a good thing.

So just for the record, tell me again:

For example you say he gives preliminary results on
zeta(s), of the form

      zeta(s) = main terms + O(error terms).

Do the _main terms_ there blow up like 1/(s-1) as s decreases to 1?

If the answer is yes, then note that in that case it _is_ obvious
that the error terms are bounded on (1,x_0). (Also _note_ that
I said the error terms are bounded on (1,x_0), not that the
O estimate is valid there.)

>but again, this is exactly my point! It isn't
>*obvious* that they don't blow up near 1; it needs
>to be *stated*. Representing the behaviour using
>the O() notation throws away vital information.
>
>(I'm aware that I've repeated myself several times
>here, and I'm sorry; but I'm getting frustrated
>trying to get my point across, and I don't know
>how many other ways there are to say it! I hope
>I haven't just muddied the waters further.)

************************

David C. Ullrich

Quantcast