# Major arcs in the proof that every odd number is the sum of at most 5 primes

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

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.
• Would there be any advantage for considering such major arcs when the generating function is $$\sum_{n\le x} e(\alpha n^2)$$ – Mayank Pandey May 12 '14 at 16:01