The second moment of Ramanujan sums

TL;DR

This paper derives an improved asymptotic formula for the second moment of Ramanujan sums under the Riemann Hypothesis, using smooth cutoff functions for flexible analysis and comparison with prior work.

Contribution

It introduces a novel method employing smooth cutoff functions to analyze the second moment of Ramanujan sums with better error terms under RH.

Findings

Asymptotic formula for $C(x, y)$ with improved error term

Uniform analysis for close values of x and y

Comparison with previous work when y=2x^2

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.