  The paper is devoted to interrelation between the zeta distribution and the
Radon transform on the space of $n \times m$ real matrices. We present a
self-contained proof of the Fourier transform formula for this distribution.
Our method differs from that of J. Faraut and A. Koranyi in the part related to
justification of the corresponding Bernstein identity. We suggest a new proof
of this identity based on explicit representation of the radial part of the
Cayley-Laplace operator. We also study convolutions with normalized zeta
distributions, and the corresponding Riesz potentials. The results are applied
to investigation of Radon transforms on the space of rectangular matrices.
