Euler Product Expression of Triple Zeta Functions

Abstract

We construct multiple zeta functions considered as absolute tensor products of usual zeta functions. We establish Euler product expressions for triple zeta functions [Formula: see text] with p, q, r distinct primes, via multiple sine functions by using the signatured Poisson summation formula. We also establish Euler product expressions for triple zeta functions [Formula: see text] with a prime p, via the theory of multiple sine functions.

The double Riemann zeta function

“…On the other hand, from Theorem 1.2 with w = 0, E (j) (s) are holomorphic in D 1 except for j = 2, 6. These imply that is also holomorphic in D 1 . From Proposition 7.1(2), the second brace is holomorphic in D 1 .…”

“…We also remark that Koyama and Kurokawa [13] obtained an Euler product expression and a Weil's-type explicit formula for ζ ⊗2 (s). Concerning the absolute tensor product for Euler factors of the Riemann zeta function, see [1][2][3][4], [11][12][13][14].…”

The Euler Product expressions of the absolute tensor products of the Dirichlet $L$-functions

“…In [1] Kurokawa also predicted that the absolute tensor product of r arithmetic zeta functions which have the expression by the Euler product over primes would have the Euler product over r-tuples (p 1 , • • • , p r ) of primes. The validity of Kurokawa's prediction has been confirmed in some cases, for example, the cases of the Hasse zeta functions of finite fields by Koyama and Kurokawa [4] for r = 2, by Akatsuka [5] for r = 3 and by Kurokawa and Wakayama [6] for general r. Also, the case of the Riemann zeta function for r = 2 was first proved by Koyama and Kurokawa [4], and then by Akatsuka [7] in a different way.…”

Absolute tensor products, Kummer's formula and functional equations for multiple Hurwitz zeta functions

“…We also expect that the (conjectural) generalized Euler product expression for the absolute tensor product of usual zeta functions Z j (s) would be related to absolute tensor products of Euler factors of Z j (s). These expectations are based on the analytic principle which is called the multiple explicit formula: see [A1,A2,KK3,KK4].…”