https://hal-mines-paristech.archives-ouvertes.fr/hal-01114315Sauvalle, BrunoBrunoSauvalleĂ‰cole des Mines de ParisOn weak Mellin transforms, second degree characters and the Riemann hypothesisHAL CCSD2015Mellin TransformRiemann HypothesisZeta function[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT]SAUVALLE, Bruno2015-02-09 16:18:182020-07-21 03:18:522015-02-09 20:57:09enPreprints, Working Papers, ...https://hal-mines-paristech.archives-ouvertes.fr/hal-01114315/documentapplication/pdf1We 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.