It can be shown that $$\sum_{a + b + c = N}\Lambda(a)\Lambda(b)\Lambda(c) = \frac{1}{2}\mathfrak{S}(N)N^2 + O(N^2\log^{-A} N)$$ for some $1\ll \mathfrak S(N)\ll 1$using the circle method. Are there any results which achieve an error term of $O(N^{2 - \delta})$ for some $\delta > 0$?
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9$\begingroup$ If there is a Siegel zero, then the main term will be affected. So this is not known. $\endgroup$– LuciaCommented Nov 7, 2017 at 5:50
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10$\begingroup$ It's not just Siegel zeros: multiplying by $e^{-N/x}$ and summing, this would give an estimate for $\sum_a \Lambda(a) e^{-a/x}$ with a power saving in the error term, which is equivalent to a zero free region of zeta of the form $\{ \mathrm{Re} s > 1-\delta \}$, which is of course open. On the other hand one does get a square root savings on GRH; see ams.org/journals/era/1997-03-15/S1079-6762-97-00031-0 $\endgroup$– Terry TaoCommented Nov 7, 2017 at 7:06
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