Twin Primes Conjecture

[R.D. Silverman]

Originally Posted by ewmayer

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

I presume that this is the Greene/Tao result that the primes contain arbitrarily
long A.P.'s?

[jinydu]

I think so.

[ewmayer]

Originally Posted by Bob Silverman

I presume that this is the Greene/Tao result that the primes contain arbitrarily
long A.P.'s?

Yes - the MathWorld link jinydu provided has a link to the preprint.

Interesting that a key part of the (alleged) proof relies on results from Goldston and Yildirim's recent work on prime gaps - even though their purported proof of a strong result about gaps proved to be fatally flawed, some good still came of it, e.g. by way of the above.

[philmoore]

I've been looking at Arenstorf's paper - it's not easy going, I've always thought analytic number theory is a tough field, but what impresses me is that the methods seem to be fairly standard ones. I haven't looked at the Turan references, perhaps Arenstorf is applying some sieving methods developed in the 1960's (my progress through the paper is quite slow, actually), but if the proof holds up, what impresses me is the persistance he has shown in putting the whole thing together with classical methods.

If you want some understanding of the Wiener-Ikehara Tauberian theorem used (but not stated) on page 21 of the paper, take a look at "Introduction to Analytic Number Theory" by K. Chandrasekharan (Springer-Verlag, 1968) where this theorem is used to prove the prime number theorem. The statement of the theorem in Chandrasekharan is as follows:

Theorem (Wiener-Ikehara). Let A(x) be a non-negative, non-decreasing function of x, defined for 0<=x< infinity. Let the integral (from zero to infinity) of A(x)*e^(-xs) dx, s=sigma+it, converge for sigma>1 to the function f(s). Let f(s) be analytic for sigma>=1, except for a simple pole at s=1 with residue 1. Then the limit as x goes to infinity of e^(-x)*A(x) is equal to 1.

It looks to me like Arensdorf uses this same theorem with x replaced by his u, and the residue and limit are equal to B_2 rather than 1.

The proof of this theorem in Chandrasekharan is a real bear! But the application of it in proving the prime number theorem is not hard.

Edwards also has a nice description of Tauberian theorems in his book "Riemann's Zeta Function".

I'll try to give Arenstorf's paper closer attention after the school term ends in a few days.

[R.D. Silverman]

Originally Posted by philmoore

I've been looking at Arenstorf's paper - it's not easy going, I've always thought analytic number theory is a tough field, but what impresses me is that the methods seem to be fairly standard ones. I haven't looked at the Turan references, perhaps Arenstorf is applying some sieving methods developed in the 1960's (my progress through the paper is quite slow, actually), but if the proof holds up, what impresses me is the persistance he has shown in putting the whole thing together with classical methods.

If you want some understanding of the Wiener-Ikehara Tauberian theorem used (but not stated) on page 21 of the paper, take a look at "Introduction to Analytic Number Theory" by K. Chandrasekharan (Springer-Verlag, 1968) where this theorem is used to prove the prime number theorem. The statement of the theorem in Chandrasekharan is as follows:

Theorem (Wiener-Ikehara). Let A(x) be a non-negative, non-decreasing function of x, defined for 0<=x< infinity. Let the integral (from zero to infinity) of A(x)*e^(-xs) dx, s=sigma+it, converge for sigma>1 to the function f(s). Let f(s) be analytic for sigma>=1, except for a simple pole at s=1 with residue 1. Then the limit as x goes to infinity of e^(-x)*A(x) is equal to 1.

It looks to me like Arensdorf uses this same theorem with x replaced by his u, and the residue and limit are equal to B_2 rather than 1.

The proof of this theorem in Chandrasekharan is a real bear! But the application of it in proving the prime number theorem is not hard.

Edwards also has a nice description of Tauberian theorems in his book "Riemann's Zeta Function".

I'll try to give Arenstorf's paper closer attention after the school term ends in a few days.

I've been looking at it in detail. AFAIK it is correct. I've been looking closely
at the proof that T(s) - B_2/(s-1) can be continued to s = 1. From that,
the theorem follows immediately from the Tauberian Thm.

What is amazing is that the entire proof (if one assumes the Tauberian thm as given) could be done for a 1st year grad course in analytic no. theory.
The proof very closely mirrors the proof of PNT and Dirichlet's Thm.

Indeed. I have questions whether the T(s) function can be replaced by
T(s) = sum( lambda(n^2+1) n^-s) [to show infinitely many primes of the
form n^2+1) or generalized to any finite linear k-tuple.....e.g.
L = Lambda
T(s) = sum(L(n-1)L(n+1)L(n+5) n^-s) with a suitable B would show infinitely
many prime triples n,n+2, n+6 etc.

[wpolly]

What about T(s)=sum(L(2^n-1) n^-s)?

[R.D. Silverman]

Originally Posted by wpolly

What about T(s)=sum(L(2^n-1) n^-s)?

I'm not sure this is a Dirchlet series. What are the characters?

[R.D. Silverman]

Originally Posted by Bob Silverman

I'm not sure this is a Dirchlet series. What are the characters?

In fact, showing that this converges anywhere would be quite difficult.
The numerators grow too fast when they are non-zero.

[ATH]

Anyone know if this proof that there exist prime arithmetic progressions of any length k also proves that there is infinitely many twin primes?

Here it says that the twin prime conjecture is a special case of the k-Tuple Conjecture:
http://mathworld.wolfram.com/k-TupleConjecture.html

But there is 2 twin prime conjectures:
http://mathworld.wolfram.com/TwinPrimeConjecture.html
and it seems the 2nd is the special case of the k-Tuple conjecture not the one that says there is infinitely many twin primes.

[ewmayer]

Originally Posted by ATH

Anyone know if this proof that there exist prime arithmetic progressions of any length k also proves that there is infinitely many twin primes?

Good question. I don't believe it does, because AFAIK the work on progressions doesn't involve restrictions on the size of the gaps between the primes in the progressions - and i'm sure if the authors had been able to extend the proof to a special case such as the twins, they would have done so.

But it is certainly to be hoped for that the approach might be able to be extended in such a fashion, even if the infinitude of the twin primes has now been established by other means (assuming no major mistakes in Arenstorf's paper - much of the detail work is beyond me, but the basic approach seems sound, since most of it makes use of existing and already proven mathematical machinery - this doesn't appear to be a case like Wiles' proof of FLT, where he developed so much new machinery that that all had to be checked, as well.)

[philmoore]

Originally Posted by Bob Silverman

I've been looking at it in detail. AFAIK it is correct. .

I assume you have probably seen this thread on the nmbrthry listserve:
http://listserv.nodak.edu/scripts/wa...406&L=nmbrthry
Just wondered if you had any insights into the questions raised by Tsz Ho Chan.

Originally Posted by Bob Silverman

I have questions whether the T(s) function can be replaced by
T(s) = sum( lambda(n^2+1) n^-s) [to show infinitely many primes of the
form n^2+1) or generalized to any finite linear k-tuple.....e.g.
L = Lambda
T(s) = sum(L(n-1)L(n+1)L(n+5) n^-s) with a suitable B would show infinitely
many prime triples n,n+2, n+6 etc.

Since the argument in Arenstorf's paper depends upon getting a closed form for his function T(s), I would guess that the first line of inquiry (for n^2+1) might be more productive than the second (for prime triples). I really haven't looked closely at it, though, and I'm still just trying to follow the details in the paper. Hopefully I can take a closer look in a few days when finals are done.