I am reading Melvyn Nathanson's book Additive Number Theory - Classical bases, and in particular getting through the proof of Vinogradov's theorem (also known as the ternary Goldbach theorem). Here the proof is due to Vaughan. In particular, to get a good bound on the contribution of the minor arc, one estimates the absolute value of $F(\alpha) = \displaystyle \sum_{p \leq N} \log(p) e(p \alpha)$ by breaking it up into three pieces via something called Vaughan's Lemma. The result is as follows:

For $u \geq 1$, let $M_u(k) = \displaystyle \sum_{d | k, d \leq u} \mu(d)$. Let $\Phi(k,l)$ be an arithmetic function of two variables. Then $\displaystyle_{u < l \leq N} \Phi(1,l) + \sum_{u < k \leq N} \sum_{u < l \leq N/k} M_u(k)\Phi(k,l) = \sum_{d \leq u} \sum_{u < l \leq N/d} \sum_{m \leq N/ld} \mu(d)\Phi(dm,l)$

I am wondering if there are other applications of this lemma. In the proof of Vinogradov's theorem, we set $u = N^{2/5}$and $\Phi(k, l) = \Lambda(l)e(\alpha k l)$ and obtain estimates on each of the pieces. In my current problem I need to improve on these estimates slightly, so I want to see how flexible Vaughan's lemma can be by looking at cases where it is used to solve other problems. Any suggestions would be welcome.

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    I remember Hugh Montgomery referring to this as "the Vaughan-mangled identity," a pun on the von Mangoldt function which appears in the identity. – Gerry Myerson Feb 13 '11 at 22:43

Vaughan's identity for the Mobuis function and averages of Kloosterman sums due to Deshouillers and Iwaniec are key components of Conrey's proof that 40% of the non-trivial zeros of the Riemann zeta-function are on the critical line.

J. B. Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line. J. Reine Angew. Math. 399 (1989), 1--26.

You may also want to look at the following two papers for other striking applications of Vaughan's identity.

H. L. Montgomery, R. C. Vaughan, "On the distribution of square-free numbers" H. Halberstam (ed.) C. Hooley (ed.) , Recent Progress in Analytic Number Theory , 1 (1981) pp. 247–256

D. R. Heath-Brown, S. J. Patterson, "The distribution of Kummer sums at prime arguments" J. Reine Angew. Math. , 310 (1979) pp. 110–130

Variants and generalizations of this kind of argument get used allllllll the time in analytic number theory. See for example the chapter "Sums over primes" in Iwaniec and Kowalski's book, or the famous paper of Heath-Brown and Patterson on the distribution of Kummer sums.

  • And that's what I get for not reading other answers first. – David Hansen Feb 13 '11 at 5:27

It would appear that a new paper by Terence Tao, "Every integer greater than 1 is the sum of at most five primes", uses Vaughan's lemma in a crucial way.

The paper is found here: http://arxiv.org/abs/1201.6656

In particular, he reformulated Vaughan's lemma as Lemma 4.12 in his paper, and earlier in the paper he mentioned that the novelty of his result is gaining much smaller constants in his bounds at the cost of a worse asymptotic bound.

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