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) ≤ c ₀ exp(Cx) and for the Laplace transform holomorphic in the disk {¦ -C¦ ≤ C} and having at least one pole of order m, we obtain lower bounds for the positive and negative parts of the measure V _± (x) > cx ^m , x > x ₀. We establish lower bounds for V _+- (x) on “short” intervals. Applications to number theory of the results obtained are considered.