References

conjectured that for k = 2, 3 and for n → ∞, n � = m k, there is an asymptotic formula for rk(n): = |{(m, p): n = m k + p, p is a prime}| and that, in particular, rk(n) → ∞ if n → ∞, n � = m k. For k ≥ 2 let Ek(X): = |{n ≤ X: rk(n) = 0} | be the number of exceptions to the “weak” conjecture rk(n) ≥ 1 for n ≥ n0(k), n not a power. We prove that there exists δ = δ(k)> 0 such that Ek(X) ≪ X 1−δ, that Ek(X + H) − Ek(X) ≪ H(log X) −A for X 7 12 (1−1/k)+ɛ ≤ H ≤ X, and also give estimates for the number of integers for which the asymptotic formula actually holds. Furthermore, we give explicit estimates for δ(k) under the Generalized Riemann Hypothesis. This paper contains in part [2, 1] and a sketch of the circle method, as used in these problems