Let $c_q\left(n\right)$ be the Ramanujan sum, i.e. $c_q\left(n\right)={\sum }_{d\left|\right(q,n)}d\mu (q/d)$, where μ is the Möbius function. In a paper of Chan and Kumchev (2012), asymptotic formulas for ${\sum }_{n\le y}{\left({\sum }_{q\le x}{c}_q\left(n\right)\right)}^k$ (k = 1,2) are obtained. As an analogous problem, we evaluate ${\sum }_{n\le y}{\left({\sum }_{n\le x}c{̂}_q\left(n\right)\right)}^k$ (k = 1,2), where $c{̂}_q\left(n\right):={\sum }_{d\left|\right(q,n)}d\left|\mu (q/d)\right|$.

Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality $|\lambda ₁p₁+\lambda ₂p²₂+\lambda ₃p{₃}^k+\varpi |\le {\left(ma{x}_j{p}_j\right)}^{-(33-29k)/\left(72k\right)+\epsilon }$ has infinitely many solutions in prime variables p₁, p₂, p₃.

For every positive integer $n$ let $p\left(n\right)$ be the largest prime number $p\le n$. Given a positive integer $n=n_1$, we study the positive integer $r=R\left(n\right)$ such that if we define recursively $n_{i+1}=n_i-p\left(n_i\right)$ for $i\ge 1$, then $n_r$ is a prime or $1$. We obtain upper bounds for $R\left(n\right)$ as well as an estimate for the set of $n$ whose $R\left(n\right)$ takes on a fixed value $k$.