On an inverse ternary Goldbach problem

Abstract

Abstract. We prove an inverse ternary Goldbach-type result. Let N be sufficiently large and c > 0 be sufficiently small. Ifcontains a composite number. This improves on the bound N 1/3+o(1) obtained in [15] using Gallagher's larger sieve. The main ingredients in our argument include a type of inverse sieve result in the larger sieve regime, and a variant of the analytic large sieve inequality.

Polynomial values modulo primes on average and sharpness of the larger sieve

“…Note that for d ≥ 3, we always have τ (d) < d. Hence it is reasonable to conjecture that Corollary 1.2 is not sharp whenever α is smaller than (and bounded away from) 1/2. See the last section in [23] for a preliminary discussion on the simplest case d = 3. Theorem 1.5 (Inverse sieve conjecture implies improved larger sieve).…”