This book is an attempt to provide an introduction to some parts, more or less important, of a subfield of elementary and analytic number theory, namely the field of arithmetical functions. There have been countless contributions to this field, but a general theory of arithmetical functions does not exist, as yet. Interesting questions which may be asked for arithmetical functions or “sequences” are, for example,

1. (1) the size of such functions,

2. (2) the behaviour in the mean,

3. (3) the local behaviour,

4. (4) algebraic properties of spaces of arithmetical functions,

5. (5) the approximability of arithmetical functions by “simpler” ones.

In this book, we are mainly concerned with questions (2), (4) and (5). In particular, we aim to present elementary and analytic results on mean-values of arithmetical functions, and to provide some insight into the connections between arithmetical functions, elements of functional analysis, and the theory of almost-periodic functions.

Of course, standard methods of number theory, such as the use of convolution arguments, Tauberian Theorems, or detailed, skilful estimates of sums over arithmetical functions are used and given in our book. But we also concentrate on some of the methods which are not so common in analytic number theory, and which, perhaps for precisely this reason, have not been refined as have the above. In respect of applications and connections with functional analysis, our book may be considered, in part, as providing special, detailed examples of well-developed theories.