Abstract
The most well-known application of Montgomery's weighted sieve is to the so-called Brun-Titchmarsh inequality, which was proved byH. L. Montgomery andR. C. Vaughan in the form π(x, k, l)≤2x(ϕ(k)log(x/k))−1 for 1≤k<x, (k, l)=1, π(x, k, l) being the number of primesp≤x andp≡l modk, ϕ(k) being Euler's function. In this paper an upper estimate is given for a certain class of two-dimensional sieve problems, among them bounds for the number of “twin primes” and the number of “Goldbach representations”.
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Siebert, H. Montgomery's weighted sieve for dimension two. Monatshefte für Mathematik 82, 327–336 (1976). https://doi.org/10.1007/BF01540603
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DOI: https://doi.org/10.1007/BF01540603