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 $S(n):={\sum }_{m_1+m_2=n}\lambda \left(m_1\right)\lambda \left(m_2\right)$, 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 $\lambda (n)$.