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Old 2010-08-16, 17:02   #1
CRGreathouse
 
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Question Is this guy for real?

http://arxiv.org/abs/1008.2381

This paper makes a fairly extravagant claim: that, on the RH, all primes satisfy
p_{n+1}-p_n<\frac{c\ \log^2p_n}{\log\log p_n}
for some c and all sufficiently large n.

In particular, this is sharper than the heuristic proposed by Cramér and its modification by Maier, and dramatically stronger than the (otherwise) best known bound on the RH,
p_{n+1}-p_n\ll\sqrt n\log n

Can someone give an opinion on this paper? I have thoughts about the proof (the key part of which is the case analysis on pp. 7-8), but I'd rather not bias anyone by speaking on particulars.


Further, can I expect the same of the other papers by the same author?
http://arxiv.org/find/math/1/au:+Car.../0/1/0/all/0/1
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Old 2010-08-16, 17:12   #2
R. Gerbicz
 
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That should be wrong, since we have a conjecture from heuristics that limsup((p(n+1)-p(n))/log(n)^2)>0. See: http://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture
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Old 2010-08-16, 17:52   #3
CRGreathouse
 
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I mentioned that conjecture and its 1990 improvement (in the constant)...
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Old 2010-08-16, 18:47   #4
R.D. Silverman
 
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Quote:
Originally Posted by CRGreathouse View Post
http://arxiv.org/abs/1008.2381

This paper makes a fairly extravagant claim: that, on the RH, all primes satisfy
p_{n+1}-p_n<\frac{c\ \log^2p_n}{\log\log p_n}
for some c and all sufficiently large n.

In particular, this is sharper than the heuristic proposed by Cramér and its modification by Maier, and dramatically stronger than the (otherwise) best known bound on the RH,
p_{n+1}-p_n\ll\sqrt n\log n

Can someone give an opinion on this paper? I have thoughts about the proof (the key part of which is the case analysis on pp. 7-8), but I'd rather not bias anyone by speaking on particulars.


Further, can I expect the same of the other papers by the same author?
http://arxiv.org/find/math/1/au:+Car.../0/1/0/all/0/1


I skimmed a couple of his papers (very superficially). They do not appear
to have been written by a crank, but it is suspicious that he does not
publish in journals; I only found arxiv and eprint references.

The claimed result above, does not "feel" right. And as you observed
it is much much stronger than RH is known to imply.
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Old 2010-08-16, 19:15   #5
CRGreathouse
 
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Quote:
Originally Posted by R.D. Silverman View Post
I skimmed a couple of his papers (very superficially). They do not appear
to have been written by a crank, but it is suspicious that he does not
publish in journals; I only found arxiv and eprint references.

The claimed result above, does not "feel" right. And as you observed
it is much much stronger than RH is known to imply.
In going through the paper I found many signs wrt the 'crank' label, some one way and some the other.

Plus: Many citations, including the important ones
Minus: No apparent mention of the major heuristics contradicted (though see the bottom of p. 1)
Plus: Outside the area of concern noted in the first post, the mathematical manipulations seem sound (if routine)
Minus: Nonprofessional typesetting -- MS Word?
Minus: Appears to prove a stronger result, that the prime gap p_{n+k}-p_n\ll\log^2p_n/\log\log p_n for any fixed k, but this is not mentioned!
Minus: Von Koch's result seems to be misstated -- I looked up the original to verify!*
Minus: Could not find affiliation (in paper or on the web), just lots of arXiv articles

On the whole it seemed shady, but not so much that I could be sure on my own so I posted here. We have lots of knowledgeable people here!


* "On the assumption that ℜ(ρ) = 1/2, it is certain that
|\pi(x)-\operatorname{Li}(x)|=O(\log x\sqrt x)"
Quote:
F(x), qui exprime combien il y a de nombres premiers < x
...
Dans l'hypothèse ℜ(ρ) = 1/2 il est certain que l'erreur commise en posant
F(x) = Li(x)
est inférieure à \log x.\sqrt x, multiplié par une constante. -von Koche, p. 182

Last fiddled with by CRGreathouse on 2010-08-16 at 19:23
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Old 2010-08-22, 17:31   #6
AntonVrba
 
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Quote:
Originally Posted by R.D. Silverman View Post
I skimmed a couple of his papers (very superficially). They do not appear
to have been written by a crank ....
Quote:
Originally Posted by CRGreathouse View Post
In going through the paper I found many signs wrt the 'crank' label, some one way and some the other....
I suppose if you been around in 1905 you would have labelled Einstein a crank; instead of being so superior and labelling a person in a public forum, it would more professional to say the paper has an error.

just my thought for the day
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Old 2010-08-22, 17:51   #7
CRGreathouse
 
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Quote:
Originally Posted by AntonVrba View Post
I suppose if you been around in 1905 you would have labelled Einstein a crank; instead of being so superior and labelling a person in a public forum, it would more professional to say the paper has an error.
My question is not whether the paper has an error or not, but rather whether the author is a crank or not. Einstein was famously wrong in his predictions in the 1935 EPR paper, but he wasn't a crank. By contrast, Don Blazys is a crank (or a very convincing troll) in his proofs of the Tijdeman-Zagier Conjecture, Fermat's Last Theorem, etc.

I have *not* come to any conclusions in regard to Carella. I'm pretty convinced that the paper I linked to is wrong, for the reasons outlined above (especially the trouble in the case arguments). Also, I dimly recall a theorem which this would seem to disprove, about the lim sup of g_n over an appropriate denominator. But I'm not sure enough of this to come to a conclusion on my own, thus my question here.

As to your particular claim: was I alive in 1905 I would not have labeled Einstein a crank. First of all, I wouldn't have been reading physics papers! Second, there was no doubt surrounding those papers, which closely followed the previous innovations in the field (by, e.g., Lorenz, Poincaré, and Minkowski). If you wanted to make this a challenge, you should have chosen 1915, where there was controversy!

Last fiddled with by CRGreathouse on 2010-08-22 at 18:12
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Old 2010-08-22, 23:02   #8
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if I under stand RH = Riemann hypothesis anyways for c to stay constant:

\frac{p_{n+1}-p_n}{(\frac{\log^2p_n}{\log\log p_n})}= c

if I did the math correct and I can't find one so far (mind you I probalby haven't searched in the proper range lol).
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Old 2010-08-23, 00:50   #9
CRGreathouse
 
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Yes, but I don't think the formula holds. That range is too small for the large gaps.

We expect (Maier 1985) something like p_{n+1}-p_n>(1-\varepsilon)C\log^2p_n infinitely often for every \varepsilon>0, with C=2e^{-\gamma}, while this says that p_{n+1}-p_n<C_1\log^2p_n/\log\log p_n which is smaller by an arbitrary amount.
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Old 2010-08-23, 11:10   #10
AntonVrba
 
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Quote:
Originally Posted by CRGreathouse View Post
Einstein was famously wrong in his predictions in the 1935 EPR paper, but he wasn't a crank. By contrast, Don Blazys is a crank (or a very convincing troll) in his proofs of the Tijdeman-Zagier Conjecture, Fermat's Last Theorem, etc.
If a person is black, white, yellow, Hindu, Christian, Muslim, genius, idiot, Harvard educated, junior high school drop out, vegetarian or alcoholic is completely immaterial, the only point of discussion is if the paper he has produced has an error or not.

Clearly Don Blazys did not prove Tijdeman-Zagier Conjecture nor Fermat's Last Theorem. However, you label and slander the person and in the same breath you claim he has proven the mentioned conjecture and theorem. Just read what you wrote (I put it in bold).

Mathematical logic only tells us that the attempted proof's are flawed or in error, but that does not prove the persons to be a crank or a convincing troll or any of the types listed in the opening sentence!

Your statements put you into exactly same league and company as the persons you label.

Last fiddled with by AntonVrba on 2010-08-23 at 11:18
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Old 2010-08-23, 12:10   #11
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Quote:
Originally Posted by AntonVrba View Post
Mathematical logic only tells us that the attempted proof's are flawed or in error, but that does not prove the persons to be a crank or a convincing troll or any of the types listed in the opening sentence!
This sounds like you are arguing that the label crank is never justified. If that is what you mean, then I disagree.
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