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Some asymptotic formulae involving Cohen-Ramanujan expansions

Arya Chandran, K Vishnu Namboothiri

Abstract

Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions $f$ and $g$ with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for $\sum_{n\leq N}f(n)g(n+h)$ where $h$ is a fixed positive integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.

Some asymptotic formulae involving Cohen-Ramanujan expansions

Abstract

Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions and with absolutely convergent Cohen-Ramanujan expansions, we derive an asymptotic formula for where is a fixed positive integer. We also provide Cohen-Ramanujan expansions for certain functions to illustrate some of the results we prove consequently.
Paper Structure (3 sections, 15 theorems, 56 equations)

This paper contains 3 sections, 15 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Notations and basic results
  3. Proofs of the Main Results

Key Result

Theorem 1.1

Suppose that $f$ and $g$ are two arithmetical functions with absolutely convergent Cohen-Ramanujan expansions $f(n) = \sum\limits_{\substack{r}}\widehat{f}(r)c_r^{(s)}(n)$ and $g(n) = \sum\limits_{\substack{k}}\widehat{g}(k)c_k^{(s)}(n)$ respectively. Suppose that $\sum\limits_{\substack{r,k}}\vert

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 17 more