Goldbach numbers represented by polynomials (Q1923687)

The author's main concern is with Goldbach type problems in short intervals over thin sets of integers. Let \(F\) be a polynomial of degree \(k\geq 1\) with integer coefficients whose leading coefficient is positive. Moreover, let \(R(n)= \sum_{r+s=n} \Lambda(r)\Lambda(s)\), where \(\Lambda\) denotes the von Mangoldt symbol, and let \[ G(n)= \prod_{p\nmid n} \Biggl(1-{1\over (p-1)^2}\Biggr)\prod_{p|n} \Biggl(1+{1\over p-1}\Biggr). \] The main result (Theorem 1) of the paper reads as follows: let \(A\), \(\varepsilon>0\) and \(N^{1/(3k)+\varepsilon}\leq H\leq N^{1/k}\). Then \[ \sum_{N^{1/k}\leq n\leq N^{1/k}+ H}|R(F(n))- F(n) G(F(N))|^2\ll_{A,\varepsilon,F} HN^2L^{-A}. \] As \(k\) can be arbitrarily large, this provides examples of thin sequences in \([N,2N]\) of cardinality \(O(N^\delta)\) with arbitrarily small positive \(\delta\), having the property that almost all their elements are Goldbach numbers. The proof is based on the method developed in the author's joint work with \textit{J. Pintz} [J. Lond. Math. Soc., II. Ser. 47, 41-49 (1993; Zbl 0806.11042)], but needs some additional ideas to overcome technical difficulties. The second theorem of the paper provides an asymptotic formula for the sum \[ \sum_{N^{1/k}\leq n\leq N^{1/k}+ H}R(F(N)), \] which is uniform for \(N^{1/(6k)+\varepsilon}\leq H\leq N^{1/k-\varepsilon}\).