Goldbach numbers in sparse sequences
It was shown by \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)] that the number of even \(n\leq N\) which are not representable as a sum of two primes is \(O(N^{1-\delta})\) with some (small) \(\delta>0\). One may ask for which ``sparse'' sequence of even numbers it is possible to show that almost all elements are Goldbach numbers (i.e. \(=p_1 +p_2)\). It was proved by \textit{A. Perelli} that this is true for the sequence of values of a fixed integer polynomial, satisfying some natural conditions [Rev. Mat. Iberoam. 12, 477-490 (1996; Zbl 0874.11069)]. In the present paper the authors consider the much sparser set of even numbers \(k\) next to the real numbers \(F(n)= \exp ((\log n)^\gamma)\) \((1< \gamma <3/2)\). They can show that the number of these \(k\)'s \((k\leq N)\) which are not Goldbach is \(\ll N\exp (-c(\log N)^{3-2 \gamma})\). The proof, which uses the Hardy-Littlewood method, requires several delicate exponential sum considerations.
scientific article; zbMATH DE number 1199188
scientific article