Tag Archives: Yitang Zhang

Friendly Introduction to Sieves with a Look Towards Progress on the Twin Primes Conjecture

This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.’ During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I’ll provide missing details and/or sources for additional reading here.

Furthermore, I like this talk, so I think it’s worth preserving.

1. Introduction

We talk about sieves and primes. Long, long ago, Euclid famously proved the infinitude of primes ($latex {\approx 300}$ B.C.). Although he didn’t show it, the stronger statement that the sum of the reciprocals of the primes diverges is true:

$latex \displaystyle \sum_{p} \frac{1}{p} \rightarrow \infty, $

where the sum is over primes.

Proof: Suppose that the sum converged. Then there is some $latex {k}$ such that

$latex \displaystyle \sum_{i = k+1}^\infty \frac{1}{p_i} < \frac{1}{2}. $

Suppose that $latex {Q := \prod_{i = 1}^k p_i}$ is the product of the primes up to $latex {p_k}$. Then the integers $latex {1 + Qn}$ are relatively prime to the primes in $latex {Q}$, and so are only made up of the primes $latex {p_{k+1}, \ldots}$. This means that

$latex \displaystyle \sum_{n = 1}^\infty \frac{1}{1+Qn} \leq \sum_{t \geq 0} \left( \sum_{i > k} \frac{1}{p_i} \right) ^t < 2, $

where the first inequality is true since all the terms on the left appear in the middle (think prime factorizations and the distributive law), and the second inequality is true because it’s bounded by the geometric series with ratio $latex {1/2}$. But by either the ratio test or by limit comparison, the sum on the left diverges (aha! Something for my math 100 students), and so we arrive at a contradiction.

Thus the sum of the reciprocals of the primes diverges. $latex \diamondsuit$

(more…)

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Recent developments in Twin Primes, Goldbach, and Open Access

It has been a busy two weeks all over the math community. Well, at least it seemed so to me. Some of my friends have defended their theses and need only to walk to receive their PhDs; I completed my topics examination, Brown’s take on an oral examination; and I’ve given a trio of math talks.

Meanwhile, there have been developments in a relative of the Twin Primes conjecture, the Goldbach conjecture, and Open Access math journals.

1. Twin Primes Conjecture

The Twin Primes Conjecture states that there are infinitely many primes $latex p$ such that $latex p+2$ is also a prime, and falls in the the more general Polignac’s Conjecture, which says that for any even $latex n$, there are infinitely many prime $latex p$ such that $latex p+n$ is also prime. This is another one of those problems that is easy to state but seems tremendously hard to solve. But recently, Dr. Yitang Zhang of the University of New Hampshire has submitted a paper to the Annals of Mathematics (one of the most respected and prestigious journals in the field). The paper is reputedly extremely clear (in contrast to other recent monumental papers in number theory, i.e. the phenomenally technical papers of Mochizuki on the ABC conjecture), and the word on the street is that it went through the entire review process in less than one month. At this time, there is no publicly available preprint, so I have not had a chance to look at the paper. But word is spreading that credible experts have already carefully reviewed the paper and found no serious flaws.

Dr. Zhang’s paper proves that there are infinitely many primes that have a corresponding prime at most $latex 70000000$ or so away. And thus in particular there is at least one number $latex k$ such that there are infinitely many primes such that both $latex p$ and $latex p+k$ are prime. I did not think that this was within the reach of current techniques. But it seems that Dr. Zhang built on top of the work of Goldston, Pintz, and Yildirim to get his result. Further, it seems that optimization of the result will occur and the difference will be brought way down from $latex 70000000$. However, as indicated by Mark Lewko on MathOverflow, this proof will probably not extend naturally to a proof of the Twin Primes conjecture itself. Optimally, it might prove the $latex p$ and $latex p+16$ – primes conjecture (which is still amazing).

One should look out for his paper in an upcoming issue of the Annals.

2. Goldbach Conjecture

I feel strangely tied to the Goldbach Conjecture, as I get far more traffic, emails, and spam concerning my previous post on an erroneous proof of Goldbach than on any other topic I’ve written about. About a year ago, I wrote briefly about progress that Dr. Harald Helfgott had made towards the 3-Goldbach Conjecture. This conjecture states that every odd integer greater than five can be written as the sum of three primes. (This is another easy to state problem that is not at all easy to approach).

One week ago, Helfgott posted a preprint to the arxiv that claims to complete his previous work and prove 3-Goldbach. Further, he uses the circle method and good old L-functions, so I feel like I should read over it more closely to learn a few things as it’s very close to my field. (Further still, he’s a Brandeis alum, and now that my wife will be a grad student at Brandeis I suppose I should include it in my umbrella of self-association). While I cannot say that I read the paper, understood it, and affirm its correctness, I can say that the method seems right for the task (related to the 10th and most subtle of Scott Aaronson’s list that I love to quote).

An interesting side bit to Helfgott’s proof is that it only works for numbers larger than $latex 10^{30}$ or so. Fortunately, he’s also given a computer proof for numbers less than than on the arxiv, along with David Platt. $latex 10^{30}$ is really, really, really big. Even that is a very slick bit.

3. FoM has opened

I care about open access. Fortunately, so do many of the big names. Two of the big attempts to create a good, strong set of open access math journals have just released their first articles. The Forum of Mathematics Sigma and Pi journals have each released a paper on algebraic and complex geometry. And they’re completely open! I don’t know what it takes for a journal to get off the ground, but I know that it starts with people reading its articles. So read up!

The two articles are

GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

and, in Pi

$p$-ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES
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