313 research outputs found
Generalising the Hardy-Littlewood Method for Primes
The Hardy-Littlewood method is a well-known technique in analytic number
theory. Among its spectacular applications are Vinogradov's 1937 result that
every sufficiently large odd number is a sum of three primes, and a related
result of Chowla and Van der Corput giving an asymptotic for the number of
3-term progressions of primes, all less than N. This article surveys recent
developments of the author and T. Tao, in which the Hardy-Littlewood method has
been generalised to obtain, for example, an asymptotic for the number of 4-term
arithmetic progressions of primes less than N.Comment: 26 pages, submitted to Proceedings of ICM 200
Linear correlations amongst numbers represented by positive definite binary quadratic forms
Given a positive definite binary quadratic form f, let r(n) = |{(x,y):
f(x,y)=n}| denote its representation function. In this paper we study linear
correlations of these functions. For example, if r_1, ..., r_k are
representation functions, we obtain an asymptotic for sum_{n,d} r_1(n) r_2(n+d)
... r_k(n+ (k-1)d).Comment: 60 pages. Small correction
On Sums of Sets of Primes with Positive Relative Density
In this paper we show that if is a subset of the primes with positive
relative density , then must have positive upper density
in
. Our argument applies the techniques developed by Green and
Green-Tao used to find arithmetic progressions in the primes, in combination
with a result on sums of subsets of the multiplicative subgroup of the integers
modulo .Comment: 21 pages, to appear in J. London Math. Soc., short remark added and
typos fixe
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