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.

ABSTRACT. This chapter deals with multiplicative arithmetical functions f, and relations between the values of these functions taken at prime powers, and the almost periodic behaviour of f. More exactly, we prove that the convergence of four series, summing the values of f at primes, respectively prime powers [with appropriate weights], implies that f is in ℬq, and (if in addition the mean-value M(f) Is supposed to be non-zero) vice versa. For this part of the proof we use an approach due to H. Delange and H. Daboussi 119761 in the special case where q = 2; the general case is reduced to this special case using the properties of spaces of almost-periodic functions obtained in Chapter VI. Finally, Daboussi's characterization of multiplicative functions in Aqwith non-empty spectrum is deduced.

INTRODUCTION

As shown in the preceding chapter, q-almost-even and q-almost-periodic functions have nice and interesting properties; for example, there are mean-value results for these functions (see VI.7) results concerning the existence of limit distributions and some results on the global behaviour of power series with almost-even coefficients. These results seem to provide sufficient motivation in the search for a, hopefully, rather simple characterization of functions belonging to the spaces Aq ⊃ Dq ⊃ ℬq of almost-periodic functions, defined in VI. 1. Of course, in number theory we look for functions having some distinguishing arithmetical properties, and the most common of these properties are additivity and multiplicativity.

PROBABILITY IN THE OLD QUANTUM THEORY

Probability was connected to quantum theory right from the start, in 1900, through the derivation of Planck's radiation law. But not much attention has been paid to the concept of probability in quantized radiation, or in the ‘quantum jumps’ from one energy level to another in an atom, and related problems. A whole chapter or book, instead of a section, could be written on the background of quantum theory in statistical mechanics and spectral analysis with this aspect in mind. Probabilistic properties were included in very many of the most important papers dealing with radiation between 1900 and 1925. It became also clear in time that Planck's law could not be derived from classical physics. Einstein admits this around 1908. Later he said, in a work of 1917, that it is a weakness of the theory of photon emission that ‘it leaves to “chance” the time and direction of the elementary processes,’ thereby, according to Pais (1982, p. 412), making explicit ‘that something was amiss with classical causality.’ It remains somewhat open how the origins and shifting interpretations of probability in the old quantum theory affected the acceptance of the probabilistic interpretation of the new quantum mechanics of 1925–1926, and of the indeterminism that found its confirmation in Heisenberg's uncertainty relation in 1927.