The second moment of Ramanujan sums

TL;DR

This work studies the second moment of Ramanujan sums in a smooth framework by introducing a rapidly decaying cutoff $W$ and applying Mellin inversion under the Riemann Hypothesis. The authors derive an RH-assuming asymptotic for the smooth second moment $C(x,y)$, with a main term proportional to $\frac{yx^2}{\zeta(2)}$ and a computable factor $\widehat{W}(1)$, plus explicitly bounded error terms that improve upon previous results especially when $y$ is near $x^2$. The approach relies on contour shifts and residue calculus applied to a double Mellin integral representation, yielding a decomposition $C(x,y)=C_1+C_2+C_3+C_4+R$ and identifying the dominant contributions from poles of zeta-functions, while showing sharper estimates in near-diagonal regimes due to the smooth cutoff. The method is flexible enough to extend to number fields and offers improved uniformity in $x$ and $y$, with potential implications for related mean-value problems involving Ramanujan sums.

Abstract

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.

Theorems & Definitions (7)

• Theorem 1.1: T.H. Chan and A.V. Kumchev
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 3.1
• Remark 3.2
• Remark 3.3