Abelian, Tauberian, and Mercerian Theorems for Arithmetic Sums

Abstract

We consider Abelian results, passing from behaviour of f to that of single or double sums as above, Tauberian theorems converse implications under additional conditions (Tauberian conditions), and Mercerian theorems, in which we pass from some comparison statement between f and a sum to a conclusion on f alone. This line of work may be traced to a seminal paper of Polya in 1917 ([P]; see also Polya & Szego [PS, Part II, Ch. 4, No. 156]). Our two main tools are the Karamata theory of regular variation (originated in 1930: see [BGT]) and the Wiener Tauberian theory (originated in 1932: see Hardy [H], Widder [W]). Polya’s achievement is all the more striking in that neither of these tools was available to him. This study arises from a fusion of two recent lines of work. On the arithmetic sums side, our interest was stimulated by a series of studies by De Koninck & Ivic [DeKI1,2,3]. On the Abelian-Tauberian-Mercerian side, we make use of recent work of our own [BI5],