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arXiv · 1502.02633

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

Bruno Sauvalle
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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 $Mell(ϕ\star f,s) = Mell(ϕ,s)M\_f(s)$ for all test functions $ϕ$ 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.

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Bruno Sauvalle. 2015-02-09. On weak Mellin transforms, second degree characters and the Riemann hypothesis. https://arxiv.org/abs/1502.02633

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