The inverse goldbach problem

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 C₁ and c₂ such that, for sufficiently large x, the following bounds hold: exp (c₁ log x/log_r x) ≪ A(x)≪ x/exp (c₂ 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.