  In this paper, Cram\'{e}r type moderate deviations for the maximum of the
periodogram and its studentized version are derived. The results are then
applied to a simultaneous testing problem in gene expression time series. It is
shown that the level of the simultaneous tests is accurate provided that the
number of genes $G$ and the sample size $n$ satisfy $G=\exp(o(n^{1/3}))$.
