Twin prime conjecture work, notation question

[eepiccolo]

First off, mad props to Dan Goldston and Cem Yildirim for their work on the twin prime conjecture! You can look for some more info at http://aimath.org/. However, I have a question about the notation in the technical description. At one point is says:

What can actually be proven about small gaps between consecutive primes? A restatement of the prime number theorem is that the average size of p_n+1-p_n is log p_n where p_n denotes the nth prime. A consequence is that
Δ := lim inf_{n -> oo} [(p_n+1-p_n) / log p_n] ≤ 1.

The thing that I don't know about is that "inf" between the "lim" and the expression. Of course, I know what a limit is and all, but I've never seen it with an inf, so could someone out there help me out? I'm really curious about this.

Thanks

[philmoore]

The symbol "inf" stands for "infimum". It is a kind of minimum value defined in a way that works for infinite sets. Suppose we have an infinite set of real numbers a_n defined for all positive integers n, and suppose that this set is bounded from below, i.e., there is some number b such that a_n > b for all n. Then it can be proven that there is a number d, called the infimum of the set {a_n| all n}, such that all a_n>=d, and if b is any other number such that a_n>=b for all n, then d>=b, so d is what we might call a "greatest lower bound". Goldston and Yildrim's theorem, then, says that there are arbitrarily large values of n such that
(p_n+1 - p_n)/(ln p_n) is arbitrarily close to zero.
We can't just take the limit of this quantity, because it's "average" size is 1, and it jumps around, but the construction "lim inf" gets around this problem.

[ewmayer]

Originally Posted by eepiccolo

First off, mad props to Dan Goldston and Cem Yildirim for their work on the twin prime conjecture!

Yes, this is very exciting work. In fact, in a recent posting (13. March) to the
number theory mailing list, Hugh Montgomery went so far as to refer to this as
"the biggest excitement that prime number theory has seen since the
Bombieri--Vinogradov theorem was proved in 1966."

The thing that I don't know about is that "inf" between the "lim" and the expression.
Of course, I know what a limit is and all, but I've never seen it with an inf, so could someone out there help me out?

It stands for "infimum", which is formally defined as the greatest lower bound of a set.
Loosely speaking, it's the value of the smallest member of a set, which must actually
be achieved for finite sets, but is defined in the sense of arbitrarily close approach for infinite sets.
Here is the relevant page from Mathworld:

http://mathworld.wolfram.com/Infimum.html

Its opposite is supremum ("sup" for short), which is the least upper bound of a set.

[eepiccolo]

Let me try to make sure I understand. Philmoore, you said that there are arbitrarily large values of n such that (p_n+1 - p_n)/(ln p_n) is arbitrarily close to zero. But does this mean it only has to happen for one n, or does (p_n+1 - p_n)/(ln p_n) get arbitrarily close to zero an infinite numebr of times?

[ewmayer]

Originally Posted by eepiccolo

Let me try to make sure I understand. Philmoore, you said that there are arbitrarily large values of n such that (p_n+1 - p_n)/(ln p_n) is arbitrarily close to zero. But does this mean it only has to happen for one n, or does (p_n+1 - p_n)/(ln p_n) get arbitrarily close to zero an infinite numebr of times?

If the quantity in question can never exactly EQUAL zero (by definition it can't in the above case, since p_n is strictly less than p_(n+1), then lim inf -> 0 implies that for any small constant epsilon, the variable in question must be less than epsilon for infinitely many values of n. That because the definition of inf implies arbitrary closeness, so if we pick a fixed (nonzero) value of epsilon, we're guaranteed to drop below it for some n. As soon as that occurs
(i.e. for the smallest n for which it does), assuming we haven't hit zero, we then pick a new epsilon which is just a bit smaller than the value we just achieved (and hence, also smaller than the old epsilon.) We're guaranteed that there's a value of n for which we'll better the new epsilon. We can repeat this (if only in our minds) ad infinitum.

[eepiccolo]

Thanks ewmayer!

[ewmayer]

News Flash - looks like the original flawed proof has been corrected, improved and extended.

From a News of the Week article title in the 27 May issue of Science by Barry Cipra titled "Third Time Proves Charm for Prime-Gap Theorem":

...now, with the help of Ja'nos Pintz of the Alfre'd Re'nyi Mathematical Institute in Budapest, Hungary, [San Jose State University's Dan] Goldston and [Turkish mathematician Cem] Yildirim have unveiled a new proof of their breakthrough result. This time experts who have examined it say the proof is rock-solid -- in part because it is much simpler than the earlier attempt.

...The original proof foundered when [the U. of Montreal's Andrew] Granville and Kannan Soundararajan of the University of Michigan, Ann Arbor, spotted a mistake in a single, technical subsection of the proof, known as a lemma. The rest of the proof was fine, and part of it immediately enabled two other mathematicians to make a major breakthrough in studying arithmetic progressions of primes (Science, 21 May 2004, p.1095). Goldston and Yildirim also salvaged a weaker result about prime gaps that improved on previous researchers' work.

Goldston kept hoping to make the proof work but finally gave up. "I had finally come to terms with not getting a good result," he recalls. Then, about a year ago, he had an idea for a new approach. He worked out the details and presented his new proof last summer at the mathematical conference center in Oberwohlfach [sic - should be "Oberwolfach"], Germany. He woke up the next morning, however, knowing he had made another mistake, this time in the very last step of the proof. "I really felt jinxed by the whole thing," he recalls.

Pintz, however, took a close look at the flawed proof and came up with the key insight for the ultimate fix. He contacted Goldston and Yildirim last December, and the three number theorists had a complete proof by early February. This time they were more cautious about announcing the result. "We all thought it was wrong," Goldston says. They circulated the manuscript to a handful of experts, including Granville and Soundararajan, asking them to probe it for any new or remaining errors.

In addition to finding nothing wrong, the ad hoc jury also discovered ways to simplify the proof. "It's been simplified so much there's not much room for an error to be hiding," says [director of the Palo Alto, California-headquartered American Institute of Mathematics' Brian] Conrey. One of the experts, Yoichi Motohashi of Nihon University in Japan, found a shortcut that led to a surprisingly short proof of the basic, qualitative result. He and the the three lead authors have posted this proof, running a mere eight pages, at the arXiv preprint server (www.arxiv.org). The more-detailed proof with Pintz is being rewritten to incorporate some of the simplifications. Goldston gave a public presentation on the new proof at a number theory conference held from 18 to 21 May at the City University of New York.

In itself, the basic result is not a surprise. But it may help mathematicians tackle the famous "twin prime" conjecture, which probably dates back as far as mathematicians have thought about prime numbers ... the new proof is still a far cry from the twin prime conjecture, but it offers a glimmer of hope that number theorists may eventually get there -- perhaps a lot sooner than they ever expected. "The twin prime conjecture doesn't seem impossible to prove anymore," Goldston says.

The AIM website, in an update posted just yesterday, adds:

The precise statement of the new theorem is that for any positive number ε there exist primes p and p' such that the difference between p and p' is smaller than ε log p. The proof of an even stronger statement, namely that the difference can be as small as (log p)^1/2 (log log p)² appears in a manuscript by the three authors that has been privately circulating; the details of this more complicated proof have not been carefully scrutinized by a large audience as of yet.

[jinydu]

Wasn't the Twin Prime Conjecture proved last year?

http://mathworld.wolfram.com/news/20...09/twinprimes/

EDIT: Just reread the last paragraph, never mind.