An Arithmetical Function Related to Báez-Duarte’s Criterion for the Riemann Hypothesis

Abstract

In this mainly expository article, we revisit some formal aspects of Báez-Duarte’s criterion for the Riemann hypothesis. In particular, starting from Weingartner’s formulation of the criterion, we define an arithmetical function ν, which is equal to the Möbius function if and only if the Riemann hypothesis is true. We record the basic properties of the Dirichlet series of ν and state a few questions.

To the memory of my friend, Luis Báez-Duarte.

Appendix: Some Computations

1.1 Scalar Products

• 1. One has

  

  (14)

• 2. For $k \in \mathbb {N}^*$, one has

  

  where

  $$\displaystyle \begin{aligned} \omega(z) &=z^{-2}\big((1-z)\ln(1-z)+(1+z)\ln(1+z)\big)-1\\ &=\sum_{j \ge 1} \frac{z^{2j}}{(j+1)(2j+1)} \quad ( \lvert z \rvert \le 1). \end{aligned} $$

• 3. For $n \in \mathbb {N}^*$, one has

  

In particular,

1.2 Projections

By (14), the orthogonal projection $e^{\prime }_1$ of e ₁ on D ₀ is

Since $e^{\prime }_1(k)$ has limit 1∕2 when k tends to infinity, one sees that $e_1-e^{\prime }_1$ “interpolates” between the fractional part (on [0, 1[ ) and the first Bernoulli function (at infinity). One has the Hilbertian decomposition

$$\displaystyle \begin{aligned} D=D_0\oplus \mathrm{Vect}(e_1-e^{\prime}_1). \end{aligned}$$

Since κ ⊥ D ₀ and , the orthogonal projection of κ on D is

$$\displaystyle \begin{aligned} \kappa'=\frac{e_1-e^{\prime}_1}{\lVert e_1-e^{\prime}_1\rVert^2}\cdotp \end{aligned}$$