### 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 |
---|---|

Pages (from-to) | 151-158 |

Number of pages | 8 |

Journal | Mathematika |

Volume | 48 |

Issue number | 1-2 |

Publication status | Published - 2001 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The inverse goldbach problem.** / Elsholtz, Christian.

Research output: Contribution to journal › Article › Research › peer-review

}

TY - JOUR

T1 - The inverse goldbach problem

AU - Elsholtz, Christian

PY - 2001

Y1 - 2001

N2 - 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 x1/2(log x)-5≪A(x)≪x1/2(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 C1 and c2 such that, for sufficiently large x, the following bounds hold: exp (c1 log x/logr x) ≪ A(x)≪ x/exp (c2 log x/logr x). (Here logr 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.

AB - 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 x1/2(log x)-5≪A(x)≪x1/2(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 C1 and c2 such that, for sufficiently large x, the following bounds hold: exp (c1 log x/logr x) ≪ A(x)≪ x/exp (c2 log x/logr x). (Here logr 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.

UR - http://www.scopus.com/inward/record.url?scp=21144454642&partnerID=8YFLogxK

M3 - Article

VL - 48

SP - 151

EP - 158

JO - Mathematika

JF - Mathematika

SN - 0025-5793

IS - 1-2

ER -