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

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Bruno Sauvalle

Résumé

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.
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Dates et versions

hal-01114315 , version 1 (09-02-2015)

Licence

Paternité - CC BY 4.0

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Bruno Sauvalle. On weak Mellin transforms, second degree characters and the Riemann hypothesis. 2015. ⟨hal-01114315⟩
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