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) $?
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asked Feb 26 '18 at 16:10
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$\begingroup$ It's certainly $O(x^2)$ :) $\endgroup$ – Wojowu Feb 26 '18 at 16:24
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$\begingroup$ And less than $ \frac{x(x-1)}{2} $! $\endgroup$ – Sylvain JULIEN Feb 26 '18 at 17:09
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6$\begingroup$ I think this question suffers from a common barrier: "assuming Goldbach's conjecture" doesn't postulate or imply any sort of quantitative information on the sizes of the primes involved. (Of course we have heuristics, but that doesn't sound like what you're looking for here.) And GRH isn't known to be of any help here. So I'd be surprised if one could do any better with GRH than without GRH. One could try to modify existing almost-all results for Goldbach's conjecture to get $o(x^2)$ for the sum you propose.... $\endgroup$ – Greg Martin Feb 26 '18 at 18:21
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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 $4/7$.
answered Feb 26 '18 at 18:57
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1$\begingroup$ GH, I didn't notice it at first look, but I think you forgot a hash sign, cause a set can't be upper bounded per se. $\endgroup$ – Sylvain JULIEN Feb 26 '18 at 21:30
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$\begingroup$ Maybe a further interesting question would be to investigate the behavior of $R_{t}(x) : =\sum_{n\leq x}r_{0}(n)^{t} $ as well as the map $ t\mapsto\frac{\partial}{\partial t}R_{t}(x) $ for $ x $ fixed. $\endgroup$ – Sylvain JULIEN Feb 26 '18 at 22:04
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$\begingroup$ @SylvainJULIEN: That's equivalent to investigating the distribution of $r_0(n)$ as a whole. $\endgroup$ – GH from MO Feb 26 '18 at 22:23
