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. 



Lower bounds on the error term of the prime number theorem
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. 


