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# Mathematics > Number Theory

# Title: Automorphic pairs of distributions on $\mathbb{R}$, and Maass forms of real weights

(Submitted on 26 Feb 2020)

Abstract: We give a correspondence between automorphic pairs of distributions on $\mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $\mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,\mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weights.

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