On sums of Ramanujan sums

Abstract

Let $c_q(n)$ denote the Ramanujan sum modulo $q$, and let $x$ and $y$ be large reals, with $x=o(y)$. We obtain asymptotic formulas for the sums $$\sum _{n\le y}(\sum _{q\le x}{c}_q(n){)}^k(k=1,2).$$