In an earlier paper (Part VII, with the same title as the present paper) we proved results on the lower bound for the number of zeros of generalised Dirichlet series $F(s)= \sum_{n=1}^{\infty} a_n\lambda^{-s}_n$ in regions of the type $\sigma\geq\frac{1}{2}-c/\log\log T$. In the present paper, the assumptions on the function $F(s)$ are more restrictive but the conclusions about the zeros are stronger in two respects: the lower bound for $\sigma$ can be taken closer to $\frac{1}{2}-C(\log\log T)^{\frac{3}{2}}(\log T)^{-\frac{1}{2}}$ and the lower bound for the number of zeros is something like $T/\log\log T$ instead of the earlier bound $>\!\!\!>T^{1-\varepsilon}$.