Tohoku Mathematical Journal
2020
September
SECOND SERIES VOL. 72, NO. 3

Tohoku Math. J.
72 (2020), 349-378

Title FUNCTIONAL EQUATIONS OF ZETA FUNCTIONS ASSOCIATED WITH HOMOGENEOUS CONES

Author Hideto Nakashima

(Received December 18, 2017, revised March 20, 2019)
Abstract. In this paper, we focus on solvable prehomogeneous vector spaces associated with homogeneous cones, and consider the associated zeta functions in several variables. We discuss $\mathbb{Q}$-structures of these prehomogeneous vector spaces, and give explicit formulas of functional equations of the zeta functions by using the data of homogeneous cones. The associated $b$-functions are also described explicitly.

Mathematics Subject Classification. Primary 11M41; Secondary 11S90, 22E25, 43A85.
Key words and phrases. Zeta functions, functional equations, $b$-functions, prehomogeneous vector spaces, homogeneous cones.

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