On a sum involving certain arithmetic functions and the integral part function

In this short, we study sums of the shape $sum_{nleqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $varphi$, Dedekind function $Psi$, sum-of-divisors function $sigma$ or the alternating sum-of-divisors function $beta.$ We improve previous results when $f=varphi$ and derive new estimates when $f=Psi, f=sigma$ and $f=beta.$

The Average Size of Ramanujan Sums over Cubic Number Fields

Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.