The inverse goldbach problem

    Research output: Contribution to journalArticleResearchpeer-review

    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 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.

    Original languageEnglish
    Pages (from-to)151-158
    Number of pages8
    JournalMathematika
    Volume48
    Issue number1-2
    Publication statusPublished - 2001

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    Sieve Methods
    Counting Function
    Ternary
    Decompose
    Sieve
    Logarithm
    Immediately
    Upper and Lower Bounds
    Denote

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    • Mathematics(all)

    Cite this

    The inverse goldbach problem. / Elsholtz, Christian.

    In: Mathematika, Vol. 48, No. 1-2, 2001, p. 151-158.

    Research output: Contribution to journalArticleResearchpeer-review

    Elsholtz, C 2001, 'The inverse goldbach problem' Mathematika, vol. 48, no. 1-2, pp. 151-158.
    Elsholtz, Christian. / The inverse goldbach problem. In: Mathematika. 2001 ; Vol. 48, No. 1-2. pp. 151-158.
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