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.
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. 


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. 

