Binary Goldbach problem with restrictions on the summands (Q1805103)

The author studies the number of Goldbach representations of even numbers in which one summand is very small. Under GRH he obtains that for any \(C_ 1>0\), \(C> 5C_ 1+ 44\) and \(H= (\log x)^ C\), for all even \(N\leq x\) with at most \(\ll x(\log x)^{-C_ 1}\) exceptions, we have \(|\{ (p,p_ 1)\): \(p+p_ 1= N\) and \(p_ 1\leq H\}| \sim \sigma(N) H/ (\log x\log H)\), where \(\sigma\) denotes the expected singular series. Moreover, there exists a thin set of primes \(\mathbb{P}_ 1\) (that is, \(| \mathbb{P}_ 1\cap [1, x]| \ll (\log x)^{C+ 1})\) such that all even \(N\leq x\) with at most \(\ll x(\log x)^{- C_ 1}\) exceptions, have \(\gg \sigma (N) (\log x)^{C -2}\) Goldbach representations \(N= p+ p_ 1\), with \(p_ 1\in \mathbb{P}_ 1\). \textit{G. Dufner} and \textit{D. Wolke} [Montash. Math. 118, 75-82 (1994; Zbl 0808.11060)] proved that, unconditionally, there exists a set \(\mathbb{P}_ 2\) with \(| \mathbb{P}_ 2\cap [1, x]|\ll (\log x)^ 2\) such that the equation \(N= p+ p_ 1\), with \(p_ 1\in \mathbb{P}_ 2\) has at least one solution for almost all even \(N\leq x\) as above. The author also gives several similar results on ternary Goldbach representations of odd integers.