Ingham showed that, assuming RH, there's an absolute constant $C > 1$ such that for any $x > 1$ the range $[x, Cx]$ contains a number $n$ such that the error term of the PNT at $n$ is positive and a number $n'$ such that the error term is negative.
Is an analogous statement for arbitrary number fields true assuming GRH?
Kaczorowski has written a few papers on this topic. One of his more recent papers gives almost this result, assuming (something somewhat weaker than) the Selberg orthogonality conjecture. The result is stated that the number of sign changes in $[1,x]$ is $\gg \log x$, which usually is deduced from a statement of the type in the OP; I didn't check if that's the case here. You could also go through this paper's references for results from past papers on the topic.
