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Old 2004-05-28, 19:25   #1
R.D. Silverman
 
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Smile Twin Primes Conjecture

In case noone has heard, Richard Arenstorf (emeritus, Vanderbilt) has
published a manuscript with a proof of the twin prime conjecture.

It is too early to say whether the paper might contain errors (I have skimmed it and probably lack the skills to determine any subtle errors).

However, the basic approach looks sound to me and represents some
serious analytic number theory.

Basically:

Let L(n) be the VonMangoldt function. Arenstorf considers the Dirichlet
series T(s) = sum(n > 3) L(n-1)L(n+1)n^-s for Re(s) > 1 and shows
that T(s) -B/(s-1) has an analytic continuation onto Re(s) = 1. This allows
use of a Tauberian Thm to show that sum [log(p) log(p+2) --> oo.
Indeed, it allows showing that the density of twin primes (the constant B)
matches the conjecture of Hardy & Littlewood.

I do not know the details of the Tauberian Thm, nor how it is applied.
[note for the uninitiated. A Tauberian Thm allows one to deduce local
behavior of a function from its long term ergodic behavior]

I will be trying to read the paper in depth this weekend. I will need to do so
with references at my side.

This is a SERIOUS piece of work.
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Old 2004-05-28, 19:34   #2
Uncwilly
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In a nut shell does it say that twin primes are endless, or what?
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Old 2004-05-28, 19:45   #3
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Quote:
Originally Posted by "Bob Silverman"
Let L(n) be the VonMangoldt function. Arenstorf considers the Dirichlet
series T(s) = sum(n > 3) L(n-1)L(n+1)n^-s for Re(s) > 1 and shows
that T(s) -B/(s-1) has an analytic continuation onto Re(s) = 1. This allows
use of a Tauberian Thm to show that sum [log(p) log(p+2) --> oo.
Ah, yes, I had rather suspected something like that.

(say WHAT??? )


But seriously, this is great news and'd put one of the most famous questions in number theory to rest. The paper can be found here. I'll skim over it but I don't expect to understand anything beyond the four lines lines of the abstract. Perhaps you'd like to give a very general summary, in layman's terms, of proof method used when you read it?

Alex
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Old 2004-05-28, 19:54   #4
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Quote:
Originally Posted by akruppa
Ah, yes, I had rather suspected something like that.

(say WHAT??? )


But seriously, this is great news and'd put one of the most famous questions in number theory to rest. The paper can be found here. I'll skim over it but I don't expect to understand anything beyond the four lines lines of the abstract. Perhaps you'd like to give a very general summary, in layman's terms, of proof method used when you read it?

Alex
I suspect that I lack the background to confirm the result in detail. I don't
know very much about Tauberian methods (except in a general way). They
allow one to deduce *local*, finite behavior of a function from its ergodic
behavior. Setting the problem up as a Dirichlet series is quite clear; this
is how one proves there are infinitely many primes in any arithmetic progression. The use of VonMangoldt's function in the series is quite clever
(but may have been obvious to one more skilled than I).

I will try to summarize after I have read it.

I am not an expert in analytic number theory. I am perhaps at the level of
the average grad student in the subject.

Bob
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Old 2004-05-28, 23:37   #5
jinydu
 
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Quote:
Originally Posted by akruppa
Ah, yes, I had rather suspected something like that.

(say WHAT??? )


But seriously, this is great news and'd put one of the most famous questions in number theory to rest. The paper can be found here. I'll skim over it but I don't expect to understand anything beyond the four lines lines of the abstract. Perhaps you'd like to give a very general summary, in layman's terms, of proof method used when you read it?

Alex
Yes, the paper claims to prove that there are infinitely many twin primes "using methods from classical analytic number theory."

Unfortunately, I can't understand much more...

Using a Google search on Arenstorf +"Twin Prime Conjecture" returned only one result:http://www.math.vanderbilt.edu/~cale.../98_11_09.html

Last fiddled with by jinydu on 2004-05-28 at 23:50
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Old 2004-05-29, 00:12   #6
ewmayer
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If true, this would be a HUGE result. Thanks for the heads-up, Bob, and thanks for the link to the manuscript, Alex.
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Old 2004-05-29, 01:09   #7
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should we search for the twins of Mersennes?
We could bag the largest twins.

Is P95 capable of doing that?

Or G or M lucas?

Does the math work to do that (check for a twin of the M's)?
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Old 2004-05-29, 01:28   #8
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Hmmm, the largest twin prime pair is tiny, only 51090 digits!
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Old 2004-05-30, 17:21   #9
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If there are infinitely many, it's only a mater of time to find them out


Luigi
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Old 2004-05-31, 02:23   #10
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On a related note, I was perusing the 21. May issue of Science this afternoon, and came across the following article in the News of the Week section, with the impressively alliterative title "Proof Promises Progress in Prime Progressions":

http://physicalsciences.ucsd.edu/new...eory052404.htm
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Old 2004-05-31, 02:33   #11
jinydu
 
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Mathworld has known about that for a while:

http://mathworld.wolfram.com/news/20...eprogressions/
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