The application of 𝜁(𝑠) to the sieve of Eratosthenes

Vinogradov, Andrey I.

doi:10.1090/trans2/013/03

Cited by 2 publications

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“…We know that this result is originally due to A. I. Vinogradov [7]. However, as has been reviewed by H. Davenport [2], the exposition of Vinogradov in [7] does not seem to be quite clear. Thus it will be worth while, we believe, to give another proof for the theorem.…”

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confidence: 90%

On the Representation of Large Even Integers as Sums of Two Almost Primes. I

Uchiyama¹

1964

Hokkaido Math. J.

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The classical Goldbach problem, which still survives unsolved, is to prove that every even integer $\geqq 6$ is a sum of two prime numbers. In 1948 A. R\'enyi [4] succeeded, by making use of his refinement of the large sieve of Yu. V. Linnik, in proving that every even integer $\geqq 6$ is a sum of a prime and of an almost prime. Here an almost prime is a positive integer $(>1)$ the total number of prime factors of which is bounded by a certain constant. Recently this result was sharpened in part by Ch.-D. Pan [3], who showed that every sufficiently large even integer can be represented as a sum of a prime and of an almost prime possessing at most five prime factors.On the other hand, A. A. Buhstab [1] has proved that every large even integer can be written as a sum of two almost primes, each of which is composed of at most four prime factors. The purpose of the present paper is to improve this result of Buh\v{s}tab. Indeed, we shall prove the following Theorem. Every sufficiently large even integer is representable as

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confidence: 90%