Lower Bounds for Positive and Negative Parts of Measures and the Arrangement of Singularities of Their Laplace Transforms

For a real measure with variation $V(x)$ satisfying the estimate $V(x)\le c_0\exp(Cx)$ and with the Laplace transform holomorphic in the disk $\{|s-C|\le C\}$ and having at least one pole of order $m$, we obtain lower bounds for the positive and negative parts of the measure $V_\pm(x)>cx^m$, $x>x_0$. We establish lower bounds for $V_\pm(x)$ on “short” intervals. Applications to number theory of the results obtained are considered.