Unconditional error term for a summatory function related to RH and Goldbach's conjecture
Granville showed under Riemann hypothesis that $ S(x) : =\sum_{n\le x}G(n)=\frac{x^2}{2}+O(x^{3/2}) $ where $ G(n)=\sum_{k_{1}+k_{2}=n}\Lambda(k_{1})\Lambda(k_{2}) $ with $ \Lambda $ the von Mangoldt function.
Gautami Bhowmik and Imre Ruzsa showed in a paper published this year untitled "Average Goldbach and the quasi-Riemann hypothesis" that an error term of the form $ O(x^{2-\delta}) $ for some $ \delta>0 $ would imply that the real parts of the non trivial zeros of Zeta are less than $ 1-\delta' $ for some $ 0<\delta'<1 $.
Do we therefore know a heuristics different from RH suggesting that the exponent of $ x $ in the error term of $ S(x) $ is indeed less than $ 2 $? Is the main term at least unconditionally proven ?
Gautami Bhowmik and Imre Ruzsa showed in a paper published this year untitled "Average Goldbach and the quasi-Riemann hypothesis" that an error term of the form $ O(x^{2-\delta}) $ for some $ \delta>0 $ would imply that the real parts of the non trivial zeros of Zeta are less than $ 1-\delta' $ for some $ 0<\delta'<1 $.
Do we therefore know a heuristics different from RH suggesting that the exponent of $ x $ in the error term of $ S(x) $ is indeed less than $ 2 $? Is the main term at least unconditionally proven ?
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