# 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.
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https://hal-mines-paristech.archives-ouvertes.fr/hal-01114315
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Submitted on : Monday, February 9, 2015 - 4:18:18 PM
Last modification on : Tuesday, July 21, 2020 - 3:18:52 AM
Long-term archiving on: : Saturday, September 12, 2015 - 9:50:33 AM

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### Identifiers

• HAL Id : hal-01114315, version 1
• ARXIV : 1502.02633

### Citation

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

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