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

Abstract

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 $Mell(\varphi \star f,s)=Mell(\varphi ,s)M\_f(s)$ for all test functions $\varphi$ in $C\_c^{\infty }(R^{\ast })$ or $C\_c^{\infty }(Q\_p^{\ast })$. 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.