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

TL;DR

This paper explores the properties of weak Mellin transforms of second degree characters on real and p-adic fields, establishing a connection between their functional equations and the Riemann hypothesis.

Contribution

It demonstrates that the weak Mellin transform of non-degenerate second degree characters satisfies a specific functional equation and links its zeros to the Riemann hypothesis.

Findings

Weak Mellin transforms satisfy a functional equation.

Zeros of the transform occur only at Re(s)=1/2.

Connection established between transforms on adeles 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.