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If you want to do research in the circle method, then my best advice is to study Vaughan's book . If as it is too dense for you, then the definite source on the subject. I have studied it by myself (as an undergraduate) and soon it helped me to write my first papers (using the circle method). I also found the circle method is not your piece to be of cakegreat use in my thesis work. You can only grow strong if you train yourself on difficult material (reading and problems). I am not saying this to be arrogant, but because this is what I was told and this is what I experienced. I make my students suffer, and it serves them well.

As to your original question: you can never say something does not work until you have proof that it is the case (I learned this wisdom from Peter Sarnak). I think one main difficulty with applying the circle method to the binary Goldbach problem is the fact that the $L^p$-norms of the generating function are too large for $p<2$. Unfortunately, I don't know a reference off-hand. Note also that the circle method is capable of showing that up to a large $x$ all but $x^{0.99}$ even numbers are a sum of two primes, and there has been recent progress on decreasing the exponent here.

EDIT: I refined my original response.

1

You should study Vaughan's book. If it is too dense for you, then the circle method is not your piece of cake. As to your original question: you can never say something does not work until you have proof that it is the case (I learned this wisdom from Peter Sarnak). I think one main difficulty with applying the circle method to the binary Goldbach problem is the fact that the $L^p$-norms of the generating function are too large for $p<2$. Unfortunately, I don't know a reference off-hand. Note also that the circle method is capable of showing that up to a large $x$ all but $x^{0.99}$ even numbers are a sum of two primes, and there has been recent progress on decreasing the exponent here.