# An arithmetical function related to B\'aez-Duarte's criterion for the Riemann hypothesis

## TL;DR

This paper explores a new arithmetical function linked to B{\'a}ez-Duarte's criterion for the Riemann hypothesis, analyzing its properties and implications for the hypothesis's validity.

## Contribution

It introduces a novel arithmetical function related to the Riemann hypothesis and examines its properties within the context of B{\'a}ez-Duarte's criterion.

## Key findings

- The arithmetical function equals the Möbius function if the Riemann hypothesis holds.
- Basic properties of the Dirichlet series of the function are established.
- Several open questions related to the function are proposed.

## Abstract

In this mainly expository article, we revisit some formal aspects of B{\'a}ez-Duarte's criterion for the Riemann hypothesis. In particular, starting from Weingartner's formulation of the criterion, we define an arithmetical function $\nu$, which is equal to the M{\"o}bius function if, and only if the Riemann hypothesis is true. We record the basic properties of the Dirichlet series of $\nu$, and state a few questions. KEYWORDS: Riemann hypothesis, arithmetical functions, Dirichlet series, Hilbert space