## Abstract

Improved upper and lower bounds of the counting functions of the conceivable additive decomposition sets of the set of primes are established. Suppose that A + B - script J ′, where script J ′ differs from the set of primes in finitely many elements only and |A|,|B|≥2. It is shown that the counting functions A(x) of A and B(x) of B, for sufficiently large x, satisfy x^{1/2}(log x)^{-5}≪A(x)≪x^{1/2}(log x)^{4}. The same bounds hold for B(x). This immediately solves the ternary inverse Goldbach problem: there is no ternary additive decomposition A + B + ℓ = script J ′, where script J ′ is as above and |A|,|B|,ℓ|≥ 2. This considerably improves upon the previously known bounds: for any r≥2, there exist positive constants C_{1} and c_{2} such that, for sufficiently large x, the following bounds hold: exp (c_{1} log x/log_{r} x) ≪ A(x)≪ x/exp (c_{2} log x/log_{r} x). (Here log_{r} x denotes the rth iterated logarithm.) The proof makes use of a combination of Montgomery's large sieve method and of Gallagher's larger sieve. This combined large sieve method may be of interest in its own right.

Original language | English |
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Pages (from-to) | 151-158 |

Number of pages | 8 |

Journal | Mathematika |

Volume | 48 |

Issue number | 1-2 |

Publication status | Published - 2001 |

## ASJC Scopus subject areas

- Mathematics(all)