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# Forum Mathematicum

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Volume 30, Issue 6

# Dirichlet series of two variables, real analytic Jacobi–Eisenstein series of matrix index, and Katok–Sarnak type result

Yoshinori Mizuno
• Corresponding author
• Graduate School of Technology, Industrial and Social Sciences, Tokushima University, 2-1, Minami-josanjima-cho, Tokushima, 770-8506, Japan
• Email
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Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/forum-2017-0119

## Abstract

We study a real analytic Jacobi–Eisenstein series of matrix index and deduce several arithmetically interesting properties. In particular, we prove the followings: (a) Its Fourier coefficients are proportional to the average values of the Eisenstein series on higher-dimensional hyperbolic space. (b) The associated Dirichlet series of two variables coincides with those of Siegel, Shintani, Peter and Ueno. This makes it possible to investigate the Dirichlet series by means of techniques from modular form.

MSC 2010: 11F50; 11F37; 11F30

Dedicated to Professor Fumihiro Sato on the occasion of his 70th birthday

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Revised: 2018-04-09

Published Online: 2018-07-12

Published in Print: 2018-11-01

Funding Source: Japan Society for the Promotion of Science

Award identifier / Grant number: 25800021

Award identifier / Grant number: 17K05175

This work is supported by JSPS Grant-in-Aid for Young Scientists (B) 25800021 and JSPS Grant-in-Aid for Scientific Research (C) 17K05175.

Citation Information: Forum Mathematicum, Volume 30, Issue 6, Pages 1437–1459, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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