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On the discrete convolution of the Liouville and Möbius functions

Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini

Abstract

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}λ\left(m_{1}\right)λ\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In particular, using the general approach introduced in a recent paper \cite{CGZ}, we will give an explicit formula for weighted averages of $S(n)$ with a general weights $f(w)$ that verify suitable conditions. This formula allows us to obtain interesting information about the Dirichlet and power series of $S(n)$ and the discrete convolution with an arbitrary numbers of factors $λ(n)$.

On the discrete convolution of the Liouville and Möbius functions

Abstract

In this article we study some properties of the discrete convolution of Liouville function , which is a Goldbach-type counting function of representations. In particular, using the general approach introduced in a recent paper \cite{CGZ}, we will give an explicit formula for weighted averages of with a general weights that verify suitable conditions. This formula allows us to obtain interesting information about the Dirichlet and power series of and the discrete convolution with an arbitrary numbers of factors .
Paper Structure (5 sections, 19 theorems, 167 equations)

This paper contains 5 sections, 19 theorems, 167 equations.

Table of Contents

  1. introduction
  2. preliminary results
  3. convolution of the summatory function L(x)
  4. the weighted averages
  5. liouville/möbius additive problem with an arbitrary number of summands

Key Result

Theorem 1

Let $\eta>0$ and let $f:\mathbb{R}\rightarrow\mathbb{C}$. Assume that: 1) $f$ has its support in $\left[a,b\right),\,a<b$, $a\in\mathbb{R},\,\eta a<1$ and $b\in\mathbb{R}\cup\left\{ +\infty\right\}$. 2) $f\in C^{1}\left(a,b\right)$. 3) $f^{\prime}$ is absolutely continuous in $\left(a,b\right)$. 4) In the case $a\eta \geq 1$, in the previous formula we have the extra term

Theorems & Definitions (28)

  • Theorem 1
  • Corollary 2
  • Lemma 3
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • Corollary 6
  • Conjecture 7
  • Lemma 8
  • ...and 18 more