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arXiv · 2603.10241

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

Marco Cantarini·Alessandro Gambini·Alessandro Zaccagnini
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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 $\lambda(n)$.

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Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini. 2026-03-10. On the discrete convolution of the Liouville and M\"obius functions. https://arxiv.org/abs/2603.10241

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