Chapter VII - The Theorems of Elliott and Daboussi

Summary

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 ℬ^q, 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 A^qwith 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 A^q ⊃ D^q ⊃ ℬ^q 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.