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On the moments of averages of Ramanujan sums

Shivani Goel, M. Ram Murty

Abstract

Chan and Kumchev studied averages of the first and second moments of Ramanujan sums. In this article, we extend this investigation by estimating the higher moments of averages of Ramanujan sums using the Brèteche Tauberian theorem. We also give a result for the moments of averages of Cohen-Ramanujan sums.

On the moments of averages of Ramanujan sums

Abstract

Chan and Kumchev studied averages of the first and second moments of Ramanujan sums. In this article, we extend this investigation by estimating the higher moments of averages of Ramanujan sums using the Brèteche Tauberian theorem. We also give a result for the moments of averages of Cohen-Ramanujan sums.
Paper Structure (8 sections, 7 theorems, 60 equations)

This paper contains 8 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction and main results
  2. The Brèteche Tauberian theorem
  3. A quick review of arithmetical functions of several variables
  4. Non-negativity of $f(n_1, ..., n_k)$
  5. Average order of $f(n_1, ..., n_k)$
  6. Higher Moments of Ramanujan sums
  7. Moments of Cohen Ramanujan sums
  8. Concluding remarks

Key Result

Theorem 1.1

For $k\ge 3$ and $y> x^k$, as $x\to \infty$, we have where $Q\in \mathbb{R}[X]$ is a polynomial of exact degree $2^k-2k-1$ and $0\le \theta\le 1.$

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Remark
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • ...and 2 more