Generating Functions Of Arithmetical functions

Abstract

Abstract The generation of arithmetical functions by means of Dirichlet series. A Dirichlet seriesis a series of the form The variable smay be real or complex, but here we shall be concerned with real values only. F(s), the sum of the series, is called the generating functionof αn. The theory of Dirichlet series, when studied seriously for its own sake, involves many delicate questions of convergence. These are mostly irrelevant here, since we are concerned primarily with the formal side of the theory; and most of our results could be proved (as we explain later in § 17.6) without the use of any theorem of analysis or even the notion of the sum of an infinite series. There are, however, some theorems which must be considered as theorems of analysis; and, even when this is not so, the reader will probably find it easier to think of the series which occur as sums in the ordinary analytical sense. We shall use the four theorems which follow. These are special cases of more general theorems which, when they occur in their proper places in the general theory, can be proved better by different methods. We confine ourselves here to what is essential for our immediate purpose.