On the Riesz Sums in Number Theory

Abstract

The Riesz means, or sometimes typical means, were introduced by M. Riesz and have been studied in connection with summability of Fourier series and of Dirichlet series [8] and [11]. In number-theoretic context, it is the Riesz sum rather than the Riesz mean that has been extensively studied. The Riesz sums appear as long as there appears the G-function. Cf. Remark 1 and [14]. As is shown below, the Riesz sum corresponds to integration while Landau's differencing is an analogue of differentiation. This integration-differentiation aspect has been the driving force of many researches on number-theoretic asymptotic formulas. Ingham's decent treatment [13] of the prime number theorem is one typical example. We state some efficient theorems that give asymptotic formulas for the sums of coefficients of the generating Dirichlet series not necessarily satisfying the functional equation.