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    $\begingroup$ While I have no idea about the particular case, I think the question for the potential of a certain new approach is usually hard to answer ... . $\endgroup$ Commented Dec 9, 2015 at 9:31
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    $\begingroup$ One definitely needs explicit constants for the applications you describe. Nevertheless, I do expect improved results for the Riemann zeta function in the long run along these lines, e.g. Bourgain himself delivered such improvements in recent years. $\endgroup$ Commented Dec 9, 2015 at 18:43
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    $\begingroup$ The short answer is that there will be no qualitative change to the error term in the prime number theorem. The best possible quantitative bounds for the mean value theorem can only change the constant in the Korobov-Vinogradov zero free region, that is the constant $c$ in $$\pi(x)=\text{li}(x)+O\left(x\exp\left(c\log(x)^{3/5}(\log \log x)^{-1/5}\right)\right).$$ $\endgroup$ Commented Dec 10, 2015 at 16:34
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    $\begingroup$ It does not directly address the question, but it's probably worth linking to this blogpost of Terry Tao: terrytao.wordpress.com/2015/12/10/… $\endgroup$ Commented Dec 11, 2015 at 17:06
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    $\begingroup$ Concerning the ZFR for the Zeta function, in his ICM talk Wooley (youtu.be/cwn6KKzX87w?t=44m19s) answers this question. This is essentially E. Naslund's comment above. $\endgroup$ Commented Dec 15, 2015 at 14:40