Under certain mild analytic assumptions one obtains a lower bound, essentially of order $r$, for the number of zeros and poles of a Dirichlet series in a disk of radius $r$. A more precise result is also obtained under more restrictive assumptions but still applying to a large class of Dirichlet series.

This paper is devoted to studying the growth and oscillation of solutions and their derivatives of higher order non-homogeneous linear differential equations with finite order meromorphic coefficients. Illustrative examples are also treated.

We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{\left(k\right)}+a_{k-1}\left(z\right)f^{(k-1)}+\cdots +a₁\left(z\right)f^{\text{'}}+D\left(z\right)f=0$, (1) where $D\left(z\right)=Q₁\left(z\right)e^{P₁\left(z\right)}+Q₂\left(z\right)e^{P₂\left(z\right)}+Q₃\left(z\right)e^{P₃\left(z\right)}$, P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z),$a_j\left(z\right)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.