What is the critical idea behind Hardy-Littlewood circle method?

Question

I want to know what the critical idea behind Hardy-Littlewood circle method is. It seems that they divide the circle into major arcs and minor arcs to ignore the singularities of generating function to be able to use tools from complex analytics to estimate the coefficients of the generating function. Is that right?

Answer 1

The generating function has no singularities in the modern treatment of the method (it is a finite exponential sum). The idea is that the generating function is small at any point which is not close to any rational number with small denominator. This reflects the expectation that the coefficients, which are number theoretic in nature, cannot correlate with a "generic" phase function. So we expect that the bulk of the integral comes from a small portion of the circle, namely the major arcs, and this expectation is confirmed when the method works (e.g. in the ternary Goldbach problem).

P.S. Of course I am talking about an integral over the circle that expresses an interesting arithmetic quantity such as the number of representations of a large odd number as a sum of three primes. The circle method is a way to estimate such a quantity by analyzing the integrand at the various points of the circle.