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.

  • $\begingroup$ Would there be any advantage for considering such major arcs when the generating function is $$\sum_{n\le x} e(\alpha n^2)$$ $\endgroup$ – Mayank Pandey May 12 '14 at 16:01
  • $\begingroup$ That depends on your application. A measure for the importance of the major arcs other than the trivial ones is the variance of the singular series. The singular series itself is usually easy to guess beforehand, so you can compare the contribution of 1 with the other major arcs. To measure the quality of the minor arc estimate compare your upper bound with an explicitly computed value. If the singular series is almost constant, and your estimate is good, you are lucky and might need only a small neighbourhood of 1. Otherwise you should include more major arcs right from the beginning. $\endgroup$ – Jan-Christoph Schlage-Puchta May 23 '14 at 17:11

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