Abel–Tauber process and asymptotic formulas

The Abel–Tauber process consists of the Abelian process of forming the Riesz sums and the subsequent Tauberian process of differencing the Riesz sums, an analogue of the integration–differentiation process. In this article, we use the Abel–Tauber process to establish an interesting asymptotic expansion for the Riesz sums of arithmetic functions with best possible error estimate. The novelty of our paper is that we incorporate the Selberg type divisor problem in this process by viewing the contour integral as part of the residual function. The novelty also lies in the uniformity of the error term in the additional parameter which varies according to the cases. Generalization of the famous Selberg Divisor problem to arithmetic progression has been made by Rieger [Zum Teilerproblem von Atle Selberg. Math. Nachr. 30 (1965), 181–192], Marcier [Sums of the form Σg(n)/f (n). Canad. Math. Bull. 24 (1981), 299–307], Nakaya [On the generalized division problem in arithmetic progressions. Sci. Rep. Kanazawa Univ. 37 (1992), 23–47] and around the same time Nowak [Sums of reciprocals of general divisor functions and the Selberg division problem, Abh. Math. Sem. Univ. Hamburg 61 (1991), 163–173] studied the related subject of reciprocals of an arithmetic function and obtained an asymptotic formula with the Vinogradov–Korobov error estimate with the main term as a finite sum of logarithmic terms. We shall also elucidate the situation surrounding these researches and illustrate our results by rich examples.