# Journal of the Ramanujan Mathematical Society

Volume 36, Issue 1, March 2021 pp. 73–84.

Representations of ax + b group and Dirichlet Series

**Authors**:
Hongyu He

**Author institution:**Department of Mathematics, Louisiana State University Baton Rouge, LA 70803, USA

**Summary: **
Let G be the ax + b group. There are essentially two irreducible infinite dimensional unitary
representations of G, (μ, L{2}(R+)) and (μ{*}, L{2}(R+)). In this paper, we give various characterizations about smooth
vectors of μ and their Mellin transforms. Let d be a linear sum of delta distributions supported on the positive
integers Z{+}. We study the Mellin transform of the matrix coefficients μ{d, f} (a) with f smooth. We express these
Mellin transforms in terms of the Dirichlet series L(s, d). We determine a sufficient condition such that the
generalized matrix coefficient μd, f is a locally integrable function and estimate the L2-norms of μ{d, f} over the Siegel
set. We further derive an inequality which may potentially be used to study the Dirichlet series L(s, d).

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