In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed to seeing. What are the advantages of using such a major arc?


Here is the paper: <http://arxiv.org/pdf/1201.6656.pdf>


I have also posted this on math.SE here: <https://math.stackexchange.com/questions/785782/major-arcs-in-the-proof-that-every-odd-number-is-the-sum-of-at-most-5-primes>



Major arcs in the proof that every odd number is the sum of at most 5 primes
The first reason is that you do not gain much by considering major arcs around rational numbers with denominator $\geq 3$. The reason is that the contribution of the major arcs $\{a/q:(a,q)=1\}$ reflects the inhomogenity of the distribution of the function in question modulo $q$, which cannot be explained by the distribution modulo proper divisors of $q$. But already $q\geq 3$ this inhomogenity is pretty small. Define $r\_q(a)$ as the number of representations of $a\pmod{q}$ as the sum of 5 residue classes coprime to $q$. Then we have $r\_3(1)=r\_3(2)=11$, $r\_3(0)=10$, and the difference of these values is pretty small. As $q$ increases, this difference decreases pretty fast.


The second reason is that a good numerical evaluation of an arc around $a/q$ requires bounds for the error term in the prime number theorem modulo $q$, and for small values (say around $10^{30}$) such estimates are surprisingly difficult.
Note that for Helfgott's proof of ternary Goldbach computations for the roots of Dirichlet $L$-series were necessary, which went way beyond everything done before in this direction. 


