The Special Values at Negative Integers of Dirichlet Series Associated with Polynomials of Several Variables

Let $P(X)$ be a product of $k$ linear forms in $r$ variables ${X_1}, \ldots ,{X_r}$ as given by \[ \begin {array}{*{20}{c}} {P({X_1}, \ldots ,{X_r}) = \prod \limits _{j = 1}^k {({a_{j1}}{X_1} + \cdots + {a_{jr}}{X_r} + {\delta _j}),} } \\ {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \operatorname {Re} {a_{ji}} > 0,\;\operatorname {Re} \left ( {{\delta _j} + \sum \limits _{i = 1}^r {{a_{ji}}} } \right ) > 0.} \\ \end {array} \] Suppose that $\beta = ({\beta _1}, \ldots ,{\beta _r})$ is an $r$-tuple of nonnegative integers. Consider the zeta function \[ Z(P,\beta )(s) = \sum \limits _{{n_{1 = 1}}}^\infty { \cdots \sum \limits _{{n_{r = 1}}}^\infty {n_1^{{\beta _1}} \cdots n_r^{{\beta _r}}P{{({n_1}, \ldots ,{n_r})}^{ - s}},\qquad \operatorname {Re} s > \frac {{r + |\beta |}} {k}} } ,\] where $|\beta | = {\beta _1} + \cdots + {\beta _r}$. $Z(P,\beta )(s)$ has an analytic continuation in the whole complex plane and it is regular at $s = 0,\; - 1,\; - 2, \ldots , - m, \ldots$. In this paper, we shall compute the explicit values of $Z(P,\beta )(s)$ at $s = 0,\; - 1,\; - 2, \ldots ,\; - m, \ldots$ and express them in terms of finite sums of polynomials in Bernoulli numbers.