Multivariate P-adic L-function

In the recent, many mathematicians studied the multiple zeta function in the complex number field. In this paper we construct the p-adic analogue of multiple zeta function which interpolates the generalized multiple Bernoulli numbers attached to χ at negative integers. §1. Introduction Let p be a fixed prime. Throughout this paper Z p , Q p , C and C p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Q p , cf.[3, 10, 17]. Let v p be the normalized exponential valuation of C p with |p| p = p −v p (p) = p −1. The Bernoulli numbers in C are defined by (1) F (t) = t e t − 1 = ∞ n=0 B n t n n! , for |t| < 2π. From the definition, one has B 0 = 1, B 1 = − 1 2 , · · ·. Also B 2k+1 = 0 for k ≥ 1. Bernoulli numbers are used to express the special values of Riemann zeta function ζ(s) = ∞ n=1 1 n s , for s ∈ C, namely ζ(2m) = (2π) 2m (−1) m−1 B 2 m 2(2m)! , m ≥ 1. For the