On weak Mellin transforms, second degree characters and the Riemann hypothesis - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année :

## On weak Mellin transforms, second degree characters and the Riemann hypothesis

Bruno Sauvalle

#### Résumé

We say that a function f defined on R or Qp has a well defined weak Mellin transform (or weak zeta integral) if there exists some function $M_f(s)$ so that we have $Mell(\phi \star f,s) = Mell(\phi,s)M_f(s)$ for all test functions $\phi$ in $C_c^\infty(R^*)$ or $C_c^\infty(Q_p^*)$. We show that if $f$ is a non degenerate second degree character on R or Qp, as defined by Weil, then the weak Mellin transform of $f$ satisfies a functional equation and cancels only for $\Re(s) = 1/2$. We then show that if $f$ is a non degenerate second degree character defined on the adele ring $A_Q$, the same statement is equivalent to the Riemann hypothesis. Various generalizations are provided.

#### Domaines

Mathématiques [math] Théorie des nombres [math.NT]

### Dates et versions

hal-01114315 , version 1 (09-02-2015)

### Licence

Paternité - CC BY 4.0

### Identifiants

• HAL Id : hal-01114315 , version 1
• ARXIV :

### Citer

Bruno Sauvalle. On weak Mellin transforms, second degree characters and the Riemann hypothesis. 2015. ⟨hal-01114315⟩

### Exporter

BibTeX TEI Dublin Core DC Terms EndNote Datacite

### Collections

402 Consultations
127 Téléchargements