The second moment of Ramanujan sums

In this paper, we study $C(x, y)$, the second moment of Ramanujan sums. Assuming the Riemann Hypothesis(RH), we establish an asymptotic formula for $C(x, y)$ with improved error term. Our analysis applies uniformly to the case where $x$ and $y$ are arbitrary close, and in particular allows for a meaningful conparison with the work of \cite{TH} in case $y=2x^2$, while keeping the computational complexity low. The method relies on the use of smooth cutoff functions, which provide greater flexibility in contour shifting.