I hope I'm not missing anything too obvious here, but I have a question on the overall setup of the circle method itself. Just recently started glossing over Vaughan's book and other sources online, and from what I understand we can write $$ R(n)=\sum_{p_1+p_2=n}1$$ as the number of ways we can write even $n$ as a sum of two primes. If $f(x)=\sum_{p} x^p$ for $p$ prime, then $f(x)^2=\sum_{n=0}^{\infty} R(n)x^n$.

Hence with a bit of complex integration, we can then analyze $R(n)$ through the perspective of integration instead: $$ R(n)=\int_0^1f(x)^2 x^{-n-1} dx = \int_{M}f(x)^2 x^{-n-1} dx + \int_{m}f(x)^2 x^{-n-1} dx \ , $$ where we now split the integration over disjoint sets called major and minor arcs. From here, I've been told that it is now ideal to have the integral over the major arcs to be large and bounded away from zero, while proving that the minor arcs contribution is negligible in comparison.

I've read ahead and apparently the circle method fails for Goldbach's conjecture due to the minor terms being of smaller order than the major arcs. This is where my confusion sets in: why does it matter if the minor arcs are negligible? If we can show that the sum of the integrals are $>1$, wouldn't that suffice to show that $R(n)>1$ (effectively showing that every even number can be written as a sum of prime numbers)? Again apologies if I'm missing something trivial here, I'm not too well versed in this stuff.


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