Understanding the Hardy-Littlewood Circle Method: An Explanation and Example

[eljose]

Could someone explain "Hardy-Littlewood" circle method?..in fact according to Wikipedia they take:

$$f(z)=\sum_{n=0}^{\infty} a(n) z^n$$

So the inverse transform to get the a(n) is:

$$2i \pi a(n)= \oint dzf(z)z^{-(n+1)}$$

This is what i understand (Don't make me look at wikipedia because the explanation is similar and there's no example.. )

The main objective of the method is supposed to get an "asymptotyc" expression for the a(n) $$a(n)\sim g(n)$$ where the function g is known but how is this done?..thanx...

 

[matt grime]

It is a simple application of the residue theorem (the integral round a closed loop is the sum of the residues). Just subsitute in the series for f, assume you can swap integral and sum and note that the integral round the unit circle of z^r is zero unless r=-1.

 

[shmoe]

That's just the residue theorem as matt has mentioned. If you take the a(n) to be 0 or 1 depending on whether n is in a set you are interested in or not (e.g. the primes), then multiplying the corresponding f's together will have coefficients that give the number of ways to write n as a sum from these sets (e.g. a sum of two primes). The radius of convergence will be 1, possibly with some singularities on the the circle of radius 1 though. The integral in your post is around a cricle of radius strictly less than 1, so things will converge absolutely and swapping the order of the sum and integral will be justified.

A basic example to try to work out is trying to find the number of ways to write n as a sum of k integers. In this case you'd take the a's to all be 1, so f(z)=1+z+z^2+... and try to find the coefficients of f^k. You can do this by a combinatorial argument of course, but it's worth working through to see how this analytical approach can be used.

You're not going to learn much about the circle method from wikipedia. A standard reference is Vaughan's "The Hardy-Littlewood Method". There are also a few introductory lecture notes available online if you care to google for them. I have a paper version by Heath-Brown that's nice, didn't find them right away online though, but you might try looking for notes by him.

 

[eljose]

-Thanks both..i usually take "Wikipedia" as first reference because is easier to understand (for example zeta regularization) at first sight.

- Also i would like to take a look at to some "introductory" paper on the subject (remember I'm not mathematician) for example at arxiv.org only as an introductory level.

-Another question could we apply "circle method" to other closed integral..in the form:

$$\oint_C dsg(x,s)$$ where g(x,s)=exp(sx) or g(x,s)=x^{-s} where the closed curve C is a semi-circle or a rectangle..thanks.

 

[shmoe]

-Thanks both..i usually take "Wikipedia" as first reference because is easier to understand (for example zeta regularization) at first sight.

Wiki won't get you very far in maths. While the math pages are generally accurate from what I've seen, they don't go into much depth at all, and you need to get some real sources if you are really interested in learning anything. The most you'll usually get from wiki are very basic definitions and hopefully some references to more in depth works.

- Also i would like to take a look at to some "introductory" paper on the subject (remember I'm not mathematician) for example at arxiv.org only as an introductory level.

Not being a mathemetician is irrelevant. The circle method is what it is, if your background is insufficient to understand the references, then improve your background.

Found these online, haven't read them but look alright (the second has some incomplete bits in the text, mostly 'broken' latex references):
http://www.math.unipr.it/~zaccagni/psfiles/didattica/HRI.pdf
http://www.math.brown.edu/~sjmiller/1/circlemethod.pdf

Here's Roger Heath-Brown's notes:
http://www.maths.ox.ac.uk/ntg/preprints/hb/montreal.pdf

You might want to check out the Vaughan reference I gave, I'm certainly no expert on the circle method, but Vaughan seems to be referenced quite frequently so starting there is probably not a bad idea. Many other texts will have some info, like Iwaniec and Kowalski's Analytic Number Theory (actually that's a source for just about anything analytic number theory related)

-Another question could we apply "circle method" to other closed integral..in the form:

$$\oint_C dsg(x,s)$$ where g(x,s)=exp(sx) or g(x,s)=x^{-s} where the closed curve C is a semi-circle or a rectangle..thanks.

um, what are you trying to do here? These integrals you've mentioned are trivial to compute with the residue theorem. The circle method is used to pick out coefficients of a series, the residue theorem tells you what integral gets the coefficient you are interested in, the work comes in trying to evaluate this integral my other means. Nothing really to do with the integrals you're asking about.