It was proved by [Rankin][1] in 1963 that
$$j(n) \geq (C+o(1) \frac{\log(x) \log_{2}(n) \log_{4}(n)}{\log^{2}_{3}(n)} $$
holds for some positive $C>0$. The value of $C$ has since been improved by [Maier and Pomerance][2] (1990) to $C=1.3125...$ and [Pintz][3] (1997) to $C=2e^{\gamma}$ (where $\gamma$ is [Euler's constant][4]).


  [1]: http://www.ams.org/mathscinet-getitem?mr=160767
  [2]: http://www.ams.org/mathscinet-getitem?mr=972703
  [3]: http://www.ams.org/mathscinet-getitem?mr=1443763
  [4]: http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant