Why Hardy-Littlewood method is called circle method? I read the phrase Hardy-Littlewood circle method in many places in Analytic number theory books and papers. I would like to know why it is called circle method, what is the idea behind it? Is it possible to visualize it?
Is there any lecture note or other reference that explains it. I read Iwaniec-Kowalski book in ANT but I did not get where the circle comes from, it says something in Cauchy integral formula, but I like to see more visualize. 
Also I did not understand exactly the meaning and reason of calling major arc and minor arc how to choose in practice which arc is major and which is minor, in this method?
Thanks for any help.
 A: It is called the circle method because, as the commentors have mentioned, the method culminates with (trying to evaluate) an integral around the unit circle in the complex plane. 
When we do this, we parameterise it using the interval [0,1]; and we split the integral over this interval into 2 parts - the major and minor arcs. The major arc is so called because it is supposed to contribute significantly more to the value of the integral than the minor arc (and hence why the latter is called minor too). Ironically the major arc actually comprises a much smaller proportion of the interval [0,1], at least when applying the circle method to the Goldbach conjecture.
To add a bit more info, we want to find an expression for the major arc and an asymptotic expression for the minor arc via big O notation. This means that as we go to infinity the minor arc contributes a negligible amount to the integral. It is apparently bounding the minor arc that is often the biggest problem in applying the method.
I found this link very helpful as an intro.
