  Normalized exponential sums are entire functions of the form
  $$
  f(z)=1+H_1e^{w_1z}+\cdots+H_ne^{w_nz},
  $$ where $H_1,\ldots, H_n\in\C$ and $0<w_1<\ldots<w_n$. It is known that the
zeros of such functions are in finitely many vertical strips $S$. The
asymptotic number of the zeros in the union of all these strips was found by R.
E. Langer already in 1931. In 1973, C. J. Moreno proved that there are zeros
arbitrarily close to any vertical line in any strip $S$, provided that
$1,w_1,\ldots,w_n$ are linearly independent over the rational numbers. In this
study the asymptotic number of zeros in each individual vertical strip is found
by relying on R. J. Backlund's lemma, which was originally used to study the
zeros of the Riemann $\zeta$-function. As a counterpart to Moreno's result, it
is shown that almost every vertical line meets at most finitely many small
discs around the zeros of $f$.
