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

2015/02/09 by Bruno Sauvalle, Sauvalle, Bruno
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Functional Equations Stability Results #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1502.02633

arxiv created 2015/02/09 · openalex publication_date 2015/02/09 · arxiv updated 2015/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⋆ f,s) = Mell(ϕ,s)M_f(s) for all test functions ϕ in C_c^∞(R^*) or C_c^∞(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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