Assume Goldbach's conjecture. Then for any large enough composite integer $ n $ $ r_{0}(n) : =\inf\{r\ge 0,(n-r,n+r)\in\mathbb{P}^{2}\} $ exists and is obviously smaller than $ n $ . Does the assumption of Chowla's conjecture imply that for all $k>0$ one has $\sup_{n\le x}\{r_{0}(n)\}=O(x^{1/k})$?

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    $\begingroup$ Why Chowla's conjecture would be of any help to estimate the distribution of Goldbach's numbers $r_0(n)$ ? I'd say the papers on the weak Goldbach conjecture and Vinogradov's theorem explain under what assumptions we can estimate $r_0(n),\max_{n \le x} r_0(n), \sum_{n \le x} r_0(n)$. $\endgroup$ – reuns Aug 14 '17 at 13:30

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