The subject of this book is probabilistic number theory. In a wide sense
probabilistic number theory is part of the analytic number theory, where
the methods and ideas of probability theory are used to study the
distribution of values of arithmetic objects. This is usually
complicated, as it is difficult to say anything about their concrete
values. This is why the following problem is usually investigated: given
some set, how often do values of an arithmetic object get into this set?
It turns out that this frequency follows strict mathematical laws. Here
we discover an analogy with quantum mechanics where it is impossible to
describe the chaotic behaviour of one particle, but that large numbers
of particles obey statistical laws. The objects of investigation of this
book are Dirichlet series, and, as the title shows, the main attention
is devoted to the Riemann zeta-function. In studying the distribution of
values of Dirichlet series the weak convergence of probability measures
on different spaces (one of the principle asymptotic probability theory
methods) is used. The application of this method was launched by H. Bohr
in the third decade of this century and it was implemented in his works
together with B. Jessen. Further development of this idea was made in
the papers of B. Jessen and A. Wintner, V. Borchsenius and B.