  We consider the oscillation properties of the remainder term $\Delta(x)$ in
the prime number formula for Beurling primes, and their relation to the
distribution of the nontrivial zeroes of the Beurling zeta function $\zeta$.
The two main directions in this study are obtaining correspondingly high
oscillation given the existence of a certain zero $\rho$ of $\zeta$ (a
direction of study initiated by Littlewood in 1937 for the Riemann zeta
function), and to describe some function theoretical connection between the
general oscillation behavior of $\Delta$ on the one hand and the general
distribution of nontrivial zeroes (a direction initiated by Phragmen and
Ingham). As a first part of our study, here we describe results about the
behavior, and in particular fine distribution of zeroes, of $\zeta$.
  The analysis here brings about some news, sometimes even for the classical
case of the Riemann zeta function. Theorem 4 provides a zero density estimate,
which is a complement to known results for the Selberg class, relying on use of
the Riemann-type functional equation, which we do not assume in the Beurling
context. We deduce a variant of a well-known theorem of Tur\'an, extending its
range of validity even for rectangles of height only h=2. We extend a zero
clustering result of Ramachandra from the Riemann zeta case. A weaker result --
which, on the other hand, is a strong sharpening of the average result from the
classic book of Montgomery -- was worked out by Diamond, Montgomery and
Vorhauer.
