Voronoï bound for a generalized divisor function

Abstract

Using hyperbola method, Dirichlet, in 1849, proved that the error term in the study of the summatory function of the divisor function d(n) is O(x) . Then in 1904, Voronoï used the method of contour integration to improve the error term as O(x13+ϵ) , for any positive ϵ . Recently, Gupta and Maji (J. Math. Anal. Appl. 507 (2022) 125738) studied the following generalized divisor function: for any k∈ N, r∈ Z , Dk,r(n)=∑dk|n(ndk)r. In this paper, we obtain a Voronoï error bound for the summatory function of Dk,r(n) . © 2023, Indian Academy of Sciences.