I've been learning about the Circle Method (at the level of the book 
"An Invitation to Modern Number Theory," by Miller and Takloo-Bighash). The arguments in the book show how the 
Circle method can be applied successfully to the ternary Goldbach problem, but fails for the binary problem (since the minor arc 
contribution can't be bounded sufficiently well). Is this the final word on the Circle method's application to this problem? Is there any hope 
that more sophisticated arguments/estimates might make it work? 


I have seen some numerical experiments that seem to indicate that the minor arc contribution is small - for example, see the undergraduate project at 
<http://web.williams.edu/go/math/sjmiller/public_html/mathlab/public_html/jr02spring/goldbach/goldbach5.pdf> .


Also, what book and/or set of notes would be good for me to continue studying the Circle Method? I have looked at 
Vaughn's book, it is too dense for me, without enough motivating material. Is the book by Nathanson ("Additive Number Theory: The Classical Bases") a 
good place to continue? 


Thanks,
Tom



The Circle Method and the binary Goldbach Problem
I would highly recommend Nathanson's book.


Also, you might want to look at the following papers for some fairly recent applications of the circle method:


Vu, Van (2000). "On a refinement of Waring's problem". Duke Mathematical Journal 105 (1).


Wooley, Trevor (2003). "On Vu's thin basis theorem in Waring's problem". Duke Mathematical Journal 120(1).


As for applying the circle method to the binary Goldbach problem, my understanding is that the minor arc estimate is a very delicate matter (see, for example, the use of Vaughan's identity to get an upper bound on the value $F(\alpha) = \sum\_{p \leq N} \log (p) e(p\alpha)$ on the minor arc. However, I believe a much more subtle point is the following quickly-glossed over fact of the proof: using Parseval's identity, we obtain the upper bound
$$\displaystyle \int\_0^1 |F(\alpha)|^2 d\alpha = \sum\_{p \leq N} (\log p)^2 \leq \log N \sum\_{p \leq N} \log p \ll N \log N$$
This coupled with the upper bound for $|F(\alpha)|$ on the minor arc is what gives the integral estimate that allows one to conclude that the contribution of the minor arc to the integral is smaller than the contribution from the major arc. Note that the analogous bound for $\displaystyle \int\_0^1 |F(\alpha)|d\alpha \ll N$ is much weaker, since one expects the contribution from the minor arc to be less than $\displaystyle \frac{N}{(\log N)^2}$. What made the ternary case work is that Parseval's identity allows a much better upper bound, and together with the estimate of $\displaystyle |F(\alpha)| \ll \frac{N}{(\log N)^{(B/2) - 4}}$ gives us a barely-good-enough bound on the minor arc contribution.


There are some concerns about the major arc estimates as well (and I believe this is a much more difficult obstacle than the minor arc, though this belief may be unfounded), but I am not able to come up with a lucid narrative at this point.


