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The easy parts of the prime number theorem September 30, 2009

Posted by David Speyer in Number theory.
5 comments

The prime number theorem states that the number of primes less than x is approximately \int_{t \leq x} dt/\log t. Proofs of the PNT tend to be easy to follow step by step, but I can never remember them once I step away from the page. I think I finally got the outline of the standard analytic proof to stick, so I’m going to record it here before I lose it:

1. Play around with \zeta(s).

Define the zeta function by

\zeta(s) = \sum_{n=1}^{\infty} n^{-s}.

Notice that \zeta(s) = \int_{t=1}^{\infty} \frac{dt}{t^s} + O(1)=1/(s-1) + O(1) as s \to 1. We have Euler’s factorization:

\zeta(s) = \prod_{p=1}^{\infty} \frac{1}{1-p^{-s}}

so

-\frac{\zeta'(s)}{\zeta(s)} = \sum_p \frac{(\log p) p^{-s}}{1- p^{-s}} = \sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s}.

Here \Lambda(n) is the Von Mangoldt function: \Lambda(n) = \log p if n=p^k and \Lambda(n)=0 otherwise. Since \zeta(s) has a first order pole at 1, we have - \zeta'(s)/\zeta(s) = 1/(s-1) + O(1). So

\sum_{n=1}^{\infty} \frac{\Lambda(n)}{n^s} = \sum_{n=1}^{\infty} \frac{1}{n^s} + O(1).

With a little care, we can see that the O(1) is an analytic function of s. So

\lim_{s \to 1} \sum_{n=1}^{\infty} \frac{\lambda(n)-1}{n^s} exists.

2. Take the limit as s goes to 1

Use the limit above to deduce that

\sum \frac{\Lambda(n)-1}{n} converges.

This is in red because it is really hard. Maybe I’ll say something about it later. But, intuitively, it makes sense.

Added in response to a comment of E. Kowalski As I discuss a little below, the fear is that \Lambda(n)-1 might behave like \sin(r \log n), for some r. In this case, we could not take the limit. If this exact problem occurred, then -\zeta'(s)/\zeta(s)+\zeta(s) would have a pole at 1+ir. A very clever trick shows that this function has no poles with real part 1. The challenge is to show that the absence of poles means that the limit is permissible.

3. Add up the partial sums

Fix \epsilon > 0. So there is an M such that, for M < w,x, we have \left| \sum_{n=w}^x \frac{\Lambda(n)-1}{n} \right| < \epsilon. Also, there must be some absolute bound C for any sum of the form \sum_{n=w}^x \frac{\Lambda(n)-1}{n}.

So

\left| \sum_{n=1}^x (\Lambda(n)-1) \right| = \left| \sum_{w=1}^x \sum_{n=w}^x \frac{\Lambda(n)-1}{n} \right| < CM + \epsilon x.

Since \epsilon is arbitrary, this shows that \sum_{n=1}^x (\Lambda(n)-1) = o(x). You are more likely to see this quoted as \sum_{n=1}^x \Lambda(n) =x +o(x).

4. Clean up

Standard techniques turn \sum_{n=1}^x (\Lambda(n)-1) = o(x) into \sum_{n=1}^x (\Lambda(n)/\log n-1/\log n) = o(x/\log x). In other words,
\pi(x) + (1/2) \pi(x^{1/2}) + \cdots = \sum_{n=1}^x (1/\log n) + o(x/\log x).
The terms \pi(x^{1/k}) are negligible for k>1.

A few notes below the fold:

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Numbers of NSF postdocs September 30, 2009

Posted by Ben Webster in Uncategorized.
8 comments

I think this past year, we were all assuming that there were more NSF postdocs, due to the stimulus, but I hadn’t heard numbers, until I decided to look them up on the NSF website. I thought you guys might be interested to see as well. These may not be perfectly accurate (I just searched the database for awards in the calendar year 200*), but they’re close enough.

2009: 56
2008: 41
2007: 29
2006: 30
2005: 32

Bleg: Windows Latex editors September 30, 2009

Posted by Ben Webster in Latex, math life.
16 comments

Masochist that I am, I’m one of the few individuals in the universe who voluntarily use all three of Windows, Mac OSX and Linux on a basically daily basis (I have Mac and Windows laptops, and use a Linux desktop in my office). Now, on Mac and Linux installations, there is no question as to which editor I will use; I confess to being an Emacs evangelist (especially to Mac users who haven’t tried Aquamacs). I recognize a lot of people have trouble with the learning curve on Emacs, but once you’re through it, every other editor just seems annoying.

Unfortunately, I’ve never found a good version of Emacs for Windows. In my day, I’ve used Winedt, and LEd (actually, I wrote much of undergrad thesis in Notepad, but I was young and foolish then), both of which I found basically fine, but I’ve always kind of felt like there must be a better program out there. Does anyone have other suggestions?

The 20 Questions Seminar September 29, 2009

Posted by Critch in 20 questions.
20 comments

The 20 questions seminar Pablo Solis and I started two weeks ago (see below) is now happening every Tuesday afternoon.  We’ll be updating the following wiki each week (on Tuesdays or Wednesdays, if we can manage) with a fresh batch of hopefully interesting questions:

http://scratchpad.wikia.com/wiki/20qs

Many generous responses from users here at the secret blogging seminar have already been incorporated there.  The wiki is free for mathematicians everywhere to post questions and answers of their own – there is a section for questions from outside the seminar.  Our inspiration is the following:

Once in a while, we’ll all come upon a question that screams “this is easy to someone else, but who?”  On the other hand, we like to hear questions that are easy for us but hard for others, so we can be useful.  So why not trade, and trade fast?  Rule #1 at the seminar is:

“1. Avoid asking questions that require a lot of background, like from your own research.  Stick to questions you think others are at least as prepared for as you are.”

When people follow it, this rule results in questions that are fun and accessible, and you often end up with answers, too.  But since usually the “poser” is the only one really interested in the answer — at least at first — we have adopted the following:

“2. Wait until the end of the seminar to discuss solutions, unless of course you can say the answer very quickly :)”

This keeps the seminar from degenerating to a dialogue about one question, which I think is the natural tendency. And just so everyone feels welcome, we have:

“3. You do not have to ask or answer a question! Just hearing what lots of people are thinking in a short time is reason enough to attend.”

I think these three ideas, someone recording the questions, and a critical mass of curious mathematicians are all that’s needed for a successful “Questions Seminar”, though time will be the only test.

In the meantime, here are last week’s questions, to be followed by next week’s (on Tuesday):
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New developments in the blnosphere September 26, 2009

Posted by Noah Snyder in blegs, blog triumphalism, Blogroll, math life.
7 comments

Let me start out by apologizing for two things, first the horrible pun in the title, and second my absence from the blog for the summer. Between moving twice (once cross-country), graduating, getting set up at a new job, buying furniture, trying to finish some papers, and being academic coordinator at Mathcamp I was pretty swamped. As a result I missed out on some developments in the math blogging.

Frequent commenter Danny Calegari started a blog in May. It pays to occasionally click on the links in comments here as sometimes you’ll find brand new blogs. My mathcamp friend, Matt Kahle, who is a postdoc at Stanford also started a blog. It has a fun mix of some elementary stuff (like the Rubik’s cube) and some of his research (which as an interesting mix of topology, combinatorics, and statistical mechanics, it definitely involves a lot of sending n to infinity in ways that would make my advisor happy). I’ve been meaning to link to both of those since sometime in June but just haven’t gotten around to it (though I did manage to add them both to the blogroll). It’s been that sort of summer, just ask me about my passport. Also, low dimensional topology has become a group blog. I find group blogging a great model both as a reader and blogger because it promotes conversations and allows one to maintain a reasonably updated blog even when someone disappears a whole summer.

Finally, over the summer there was a great conversation about what mathematicians need to know about blogging. Here’s my two cents. One thing incredibly valuable thing about blogging is the opportunity to have discussions and get advice about how to be a mathematician. It’s often hard in real life to have a discussion involving people at many different places in their careers about professional questions. In that spirit, here’s a question I’ve been wrestling with lately. How do you balance your research time between the following three activities: working on problems you basically know how to solve, working on problems you don’t know how to solve but are important problems, and learning new tools. When I was in graduate school I felt like it was pretty easy to balance things because any time I had any idea that was at all worthwhile I just worked on it and when I didn’t, I learned new things. I had few enough research-worthy ideas that it was feasible to think about all of them. Now that I know more I can’t keep doing that because I simply don’t have time to work on all the easy problems that I could solve. So the need comes to prioritize. I was wondering how other people strike this balance.

Logicomix: An Epic Search For Graphic Novels September 25, 2009

Posted by Ben Webster in Uncategorized.
4 comments

On a lighter note, I’m informed that there’s recently been a graphic novel published on the life of Bertrand Russell. I haven’t read it myself, but LogBlog likes it. I’m of a mixed opinion about Russell’s achievements as a mathematician, but no one can deny he had an interesting life. Of particular interest to some readers is the on-going tour, coming (maybe) to a town near yours soon.

A (partial) explanation of the fundamental lemma and Ngo’s proof September 24, 2009

Posted by Joel Kamnitzer in Algebraic Geometry, geometric Langlands, Number theory, representation theory, things I don't understand.
4 comments

I would like to take Ben up on his challenge (especially since he seems to have solved the problem that I’ve been working on for the past four years) and try to explain something about the Fundamental Lemma and Ngo’s proof.  In doing so, I am aided by a two expository talks I’ve been to on the subject — by Laumon last year and by Arthur this week.

Before I begin, I should say that I am not an expert in this subject, so please don’t take what I write here too seriously and feel free to correct me in the comments.  Fortunately for me, even though the Fundamental Lemma is a statement about p-adic harmonic analysis, its proof involves objects that are much more familiar to me (and to Ben).  As we shall see, it involves understanding the summands occurring in a particular application of the decomposition theorem in perverse sheaves and then applying trace of Frobenius (stay tuned until the end for that!).

First of all I should begin with the notion of “endoscopy”.  Let G, G' be two reductive groups and let \hat{G}, \hat{G}' be there Langlands duals.  Then G' is called an endoscopic group for G if \hat{G}' is the fixed point subgroup of an automorphism of \hat{G} .  A good example of this is to take G = GL_{2n} , G' = SO_{2n+1} .  At first glance these groups having nothing to do with each other, but you can see they are endoscopic since their dual groups are GL_{2n} and Sp_{2n} and we have Sp_{2n} \hookrightarrow GL_{2n} .

As part of a more general conjecture called Langlands functoriality, we would like to relate the automorphic representations of G to the automorphic representations of all possible endoscopic groups G' .  Ngo’s proof of the Fundamental Lemma completes the proof of this relationship.

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A hunka hunka burnin’ knot homology September 24, 2009

Posted by Ben Webster in category O, Category Theory, combinatorics, homological algebra, link homology, low-dimensional topology, quantum groups, representation theory.
19 comments

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). (more…)

The Jellyfish Algorithm September 24, 2009

Posted by Noah Snyder in Paper Advertisement, planar algebras, quantum topology.
1 comment so far

Stephen Bigelow, Scott Morrison, Emily Peters and I have a preprint up on the arxiv today about the extended Haagerup subfactor and its planar algebra. Scott already explained nicely the story that this paper fits into. Today I wanted to tell you about one of the key elements of this paper, Stephen’s “Jellyfish algorithm.” This algorithm is a kind of “evaluation algorithm” and I hope to get around to putting up some more posts on some other evaluation algorithms as well as some open questions concerning evaluation algorithms (one would prove the 4-color theorem and another is related to whether the exceptional lie algebras aren’t so exceptional after all).

What is an evaluation algorithm? Well suppose you have a bunch of diagrams (think formal sums of links) constructed out of a few generators (think crossings) modulo some relations (think the HOMFLY relations). There are two obvious questions. Are these rules enough to reduce every link to a constant times the empty diagram? Does any way of reducing a given diagram give the same answer? Usually the former question is substantially easier than the latter. Often the first question is answered by giving an explicit algorithm for simplifying diagrams.

One simple example is the Kauffman bracket description of the Jones polynomial. Here the evaluation algorithm is very easy: find a crossing and resolve it, wash, rinse, repeat. This clearly terminates because at every step you get fewer crossings.
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New conferences in the sidebar September 23, 2009

Posted by David Speyer in Uncategorized.
3 comments

I don’t know how many of you noticed, but we have a category in the sidebar for conferences that have caught our eye. We’ve let it get out of date — for the last few months, all the conferences listed had already happened! I just went through and cleaned out all the past events, and added a bunch of fresh new conferences that I like the look of. I encourage my co-bloggers to add recommendations of their own; and our readers to add recommendations in the comments.

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