On the zeros of a class of generalised Dirichlet series-VIII

International audienceIn 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 $>\!\!\!>T^{1-\varepsilon}$