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  • $s=1$ line gives: $$\psi(x) = x(1+o(1))$$

  • classical zero free region gives: $$\psi(x) = x + O(x e^{-c \sqrt{\log x}})$$ for some positive constant $\delta$

  • RH gives: $$\psi(x) = x + O(\sqrt{x}\log(x)^2)$$

I hope these are correct, https://terrytao.wordpress.com/2009/09/24/the-prime-number-theorem-in-arithmetic-progressions-and-dueling-conspiracies/

So I was wondering what error term would we get if someone proved there were no zeros with real part $< 1-\varepsilon$?

and why does Terry Tao say understanding the error term is so important?

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I have seen this type of assumption referred to as a "quasi-Riemann hypothesis". See e.g. p. 5 of homepages.math.uic.edu/~cojocaru/twin-primes-march05.pdf. (Although it looks weaker, it is not clear that it is any more tractable than RH itself!) –  Pete L. Clark Feb 16 '13 at 19:50
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up vote 2 down vote accepted

Instead of the exponent $1/2$ as in the $\sqrt x$ in the error, you get the larger $x^{1-\epsilon}$

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