Assuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound for $ R(x) : =\sum_{n\leq x}r_{0}(n) $?
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Sign up to join this communityAssuming Goldbach's conjecture, let's define for a sufficiently large integer $ n $ the quantity $ r_{0}(n) : =\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\} $.
Under GRH, what is the best upper bound for $ R(x) : =\sum_{n\leq x}r_{0}(n) $?
By a result of Jia (Three primes theorem in a short interval. VII., Acta Math. Sinica (N.S.) 10 (1994), 369-387), we have for any $\epsilon>0$, $$\#\{n\leq x:\ r_0(n)>x^{7/12+\epsilon}\}\ll\frac{x}{\log^2 x}. $$ As a consequence, we have the uniform bound (under the Goldbach conjecture) $$R(x)\ll\frac{x^2}{\log^2 x}.$$ I believe that with current technology one can also establish, for some small but effective $\delta>0$, $$\#\{n\leq x:\ r_0(n)>x^{1-\delta}\}\ll x^{1-\delta}.$$ This would yield (under the Goldbach conjecture) the stronger bound $R(x)\ll x^{2-\delta}$.
Added. Baker and Harman (The three primes theorem with almost equal summands, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1738, 763–780) improved the exponent $7/12+\epsilon$ above to $4/7$.