On the discrete convolution of the Liouville and M\"obius functions

TL;DR

This paper investigates properties of a Goldbach-type counting function formed by the discrete convolution of Liouville functions, providing explicit formulas for weighted averages and insights into associated series.

Contribution

It introduces a general explicit formula for weighted averages of the convolution of Liouville functions, extending understanding of their Dirichlet and power series.

Findings

Derived explicit formulas for weighted averages of the convolution function

Analyzed the Dirichlet series associated with the convolution

Extended results to convolutions with multiple factors

Abstract

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$, 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 $\lambda(n)$.