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 $Mell(\phi \star f,s) = Mell(\phi,s)M_f(s)$ 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.