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Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.

$$f(x)\ll |\psi(x) - x|$$

where $\psi$ is the Chebyshev function.

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Just to clarify: Lucia's answer does not mean that $|\psi(x)-x|$ is always large, but only that it is large along some particular values $x$ tending to infinity. As $\psi(x)-x$ changes sign infinitely often, it is also very small (namely $\ll\log x$) along some particular values $x$ tending to infinity. –  GH from MO Aug 16 at 6:13

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Littlewood proved that $$ \psi(x)-x = \Omega_{\pm}(x^{\frac 12} \log \log \log x); $$ here $\Omega_{\pm}$ means that the LHS infinitely often gets as large as a positive constant times the RHS, and also infinitely often as small as a negative constant times the RHS. Montgomery conjectured that $$ \limsup_{x\to \infty} \frac{\psi(x)-x}{\sqrt{x}(\log \log \log x)^2} = \frac 1{2\pi}, $$ and $$ \liminf_{x\to \infty} \frac{\psi(x)-x}{\sqrt{x} (\log \log \log x)^2} = -\frac{1}{2\pi}. $$ This conjecture appears in Montgomery's paper The zeta function and prime numbers (Proceedings of the Queen's Number theory conference, 1979). A discussion of such results may be found in Chapter 15 of Montgomery and Vaughan's book Multiplicative Number Theory I. Classical Theory.

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Did Montgomery conjecture a distribution function for the $(\psi(x)-x)/\sqrt x(\log\log\log x)^2$, or just liminfsup values? –  NAME_IN_CAPS Aug 16 at 7:35
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The distribution of $(\psi(e^t)-e^t)/e^{t/2}$ is conjectured to be a certain non-universal distribution arising from the zeros of $\zeta(s)$. Assuming RH and the linear independence of ordinates of zeros of $\zeta$, one can establish this distribution function. Monach, a student of Montgomery, computed this distribution function numerically in his thesis. Montgomery found upper and lower bounds for the tails, and this was part of his motivation for the conjecture. See Rubinstein and Sarnak's paper Chebyshev's bias for closely related material. –  Lucia Aug 16 at 16:09

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