Voronoi bound for a generalized divisor function

Abstract

Let d(n) be the well-known divisor function. Using hyperbola method, Dirichlet, in 1849, proved that X nx d(n) = x log x + (2) with E(x) = O( px). After a long period of time, in 1904, Voronoi used the method of contour integration to improve the error term as O ⇣ x 1 3 +✏ ⌘ , for any positive ✏. Recently, Gupta and Maji studied a generalized form of d(n) given by Dk,r(n) := X dk|n ✓ n dk ◆r where k 2 N and r 2 Z. In this thesis, we study the summatory function of Dk,r(n) and establish a Voronoi-type bound for the error term. Moreover, we recover the Voronoi’s error bound for the summartory function of d(n)