Riemann Hypothesis?

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G. A. Edgar

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Jan 3, 2008, 8:00:39 AM1/3/08
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Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
Hypothesis.
arXiv:0801.0114v1
It is 6 pages long.
Comments?

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/

Gus Gassmann

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Jan 3, 2008, 8:25:28 AM1/3/08
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On Jan 3, 1:00 pm, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
wrote:

This is not my area, and I have not read the paper very carefully.
However, I would be weary, given this remark on page 5:

"Since the proof of [Aizenberg's] Theorem 2.1 is brief, elementary and
pretty simple and I see no mistakes in it, while the proof of [a
reference that contradicts Theorem 2.1] is complicated and rests on
four lemmas, all quite non-elementary, I suspect that [the other
reference] is incorrect."

That sounds a bit like AP claiming that FLT must be wrong because he
has a very simple and elementary result that contradicts FLT, while
Wiles' proof is long and complicated.

Also, the way I understand it, the Riemann hypothesis says that all
(nontrivial) zeroes of the zeta function lie on the line s = 1/2, so I
would expect a theorem contradicting this to exhibit a nontrivial zero
off that line. All we get is an asymptotic analysis, which at best can
show that zeta(s+it) tends to zero as t tends to infinity. This proof
technique again reminds me of AP.

Phil Carmody

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Jan 3, 2008, 8:27:34 AM1/3/08
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"G. A. Edgar" <ed...@math.ohio-state.edu.invalid> writes:
> Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
> Hypothesis.
> arXiv:0801.0114v1
> It is 6 pages long.
> Comments?

It's not in the 'GM' section, so it's at least twice
as likely to be correct as the papers therein.

Phil
--
Dear aunt, let's set so double the killer delete select all.
-- Microsoft voice recognition live demonstration

David Bernier

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Jan 3, 2008, 10:32:51 AM1/3/08
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G. A. Edgar wrote:
> Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
> Hypothesis.
> arXiv:0801.0114v1
> It is 6 pages long.
> Comments?

In Remark 2.1, the author writes that Theorem 8.12 in [10]
contradicts Theorem 2.1 in his arXiv paper,
where Reference [10] is as follows:

[10] Titchmarsh E.C.,
``The Theory of Riemann zeta-function" , Oxford Press 1988.

-----

In < ftp://ftp.esi.ac.at/pub/Zetaproc/conrey.pdf >,

J.B. Conrey mentions a lower bound for
max_{t in [0, T]} |zeta(1/2+i*t)| due to
Balasubramanian and Ramachandra, and published in 1977 in
the Proc. Indian Acad. Sci. ( Note: Conrey uses >> and
<< as part of bound results. Maybe this would mean
"for sufficiently large T" ? I don't know.)

David Bernier

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quasi

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Jan 3, 2008, 1:24:27 PM1/3/08
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On Thu, 03 Jan 2008 08:00:39 -0500, "G. A. Edgar"
<ed...@math.ohio-state.edu.invalid> wrote:

>
>Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
>Hypothesis.
>arXiv:0801.0114v1

For those not familiar with the arXiv, here is a link:

http://arxiv.org/abs/0801.0114v1

Gc

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Jan 3, 2008, 3:51:07 PM1/3/08
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On 3 tammi, 15:00, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
wrote:


I am no expert, and been looking in basics of the Riemann hypothesis
only on a couple of month`s now and then. I wonder when in the article
Aizenberg says that the relation (bigO)(t^epsilon) = zeta(1/2 + it) is
equivalent to lim t-->oo|zeta(1/2 + it|/|t|^e = 0. To me that seems
more like a small o, but I haven`t played with this notations at all,
so I may be wrong.

Gc

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Jan 3, 2008, 3:57:26 PM1/3/08
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quasi

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Jan 3, 2008, 6:47:48 PM1/3/08
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To defend his claim, it would be nice if he could produce a zero off
the critical line, giving the actual components, at least
approximately.

quasi

José Carlos Santos

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Jan 3, 2008, 8:00:45 PM1/3/08
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On 03-01-2008 23:47, quasi wrote:

>>> Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
>>> Hypothesis.
>>> arXiv:0801.0114v1
>> For those not familiar with the arXiv, here is a link:
>>
>> http://arxiv.org/abs/0801.0114v1
>>
>>> It is 6 pages long.
>>> Comments?
>
> To defend his claim, it would be nice if he could produce a zero off
> the critical line, giving the actual components, at least
> approximately.

Yes, it would be nice, but not really necessary. In 1914, Littlewood
proved that the assertion pi(x) <= Li(x) (where _pi_ is the
prime-counting function and _Li_ is the logarithmic integral function)
is not always true. However, nobody has ever provided an example of a
number _x_ such that pi(x) > Li(x).

Best regards,

Jose Carlos Santos

David C. Ullrich

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Jan 4, 2008, 7:39:17 AM1/4/08
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On Thu, 3 Jan 2008 12:57:26 -0800 (PST), Gc <Gcu...@hotmail.com>
wrote:

Erm, thanks. We know what O and o mean.

You need to read at least a tiny amount of the text
surrounding the formulas. Saying that for _every_ e > 0

f(t) = O(t^e) (t -> infinity)

is obviously equivalent to saying that

f(t) = o(t^e) (t -> infinity)

for every e > 0.


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

David C. Ullrich

Timothy Murphy

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Jan 4, 2008, 8:02:01 AM1/4/08
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David C. Ullrich wrote:

>>> > Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
>>> > Hypothesis.
>>> > arXiv:0801.0114v1
>>> > It is 6 pages long.

> Erm, thanks. We know what O and o mean.

But it did strike me that a person who takes 20% of a paper
to explain the O/o notation
is unlikely to have disproved the Riemann Hypothesis.

A bit like a proof of FLT that starts by explaining
what a prime number is.

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland

Oleg Eroshkin

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Jan 4, 2008, 12:41:24 PM1/4/08
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Timothy Murphy wrote:

>
> But it did strike me that a person who takes 20% of a paper
> to explain the O/o notation
> is unlikely to have disproved the Riemann Hypothesis.
>

Well the author's mistake is very simple. On page 3, he bounds |I_1| by
something + Integral of (t-t0) phi(t) f(t),
where f(t) is some positive function and phi(t) < C t.
However (t-t0) changes sign on the interval of integration, so it does not
follow, that this integral is bounded by integral of C(t-t0) t f(t).

--
Oleg Eroshkin
olegeroshkin (at) gmail (dot) com

A N Niel

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Jan 4, 2008, 3:38:38 PM1/4/08
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In article <fllr45$ik8$1...@tabloid.unh.edu>, Oleg Eroshkin
<firstnam...@gmail.com> wrote:

This error seems to be the consensus.

So we have an illustration of the difference between a preprint
(such as in arXiv) and a paper published in a refereed journal.
Because of the established process in mathematics, errors in the
latter type of paper are far less frequent than in the former.

David C. Ullrich

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Jan 5, 2008, 8:03:49 AM1/5/08
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On Fri, 04 Jan 2008 14:02:01 +0100, Timothy Murphy
<t...@birdsnest.maths.tcd.ie> wrote:

>David C. Ullrich wrote:
>
>>>> > Lev Aizenberg (Bar Ilan University) claims invalidity of Riemann
>>>> > Hypothesis.
>>>> > arXiv:0801.0114v1
>>>> > It is 6 pages long.
>> Erm, thanks. We know what O and o mean.
>
>But it did strike me that a person who takes 20% of a paper
>to explain the O/o notation
>is unlikely to have disproved the Riemann Hypothesis.
>
>A bit like a proof of FLT that starts by explaining
>what a prime number is.

Well right. I wasn't meaning to say anything about what
the error _was_, just pointing out that the supposed
error Gc gave was not erroneous at all.


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

David C. Ullrich

H. Shinya

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Jan 7, 2008, 12:59:28 AM1/7/08
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> "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
[the first-order > comment was written by Phil; not by G. A. Edgar.]

> writes:
> > Lev Aizenberg (Bar Ilan University) claims
> invalidity of Riemann
> > Hypothesis.
> > arXiv:0801.0114v1
> > It is 6 pages long.
> > Comments?
>
> It's not in the 'GM' section, so it's at least twice
> as likely to be correct as the papers therein.

A wrong paper is wrong; there is no degree of correctness. Therefore, in general, if a paper is about a proof of the Riemann Hypothesis, then credibility of the author for that paper is merely that he has posted a paper in the NT section.

If there had been no arXiv, Pereleman would have submitted his paper on Poincare conjecture to a peer-reviewed journal, and I would never have made ridiculous mistakes there.

> ...
> Phil

H. Shinya

H. Shinya

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Jan 8, 2008, 2:54:05 AM1/8/08
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Correction to the statement above, although no one may care. After all, I tried to spread my papers via the free homepage tool. That means arXiv can not be that bad to me.

> ...

H. Shinya

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