Barnes zeta function

In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function.

Definition

The Barnes zeta function is defined by

${\zeta }_N(s,w\mid a_1,\dots ,a_N)=\sum _{n_1,\dots ,n_N\ge 0}\frac{1}{(w+n_1{a}_1+\cdots +n_N{a}_N{)}^s}$

where w and a_j have positive real part and s has real part greater than N.

It has a meromorphic continuation to all complex s, whose only singularities are simple poles at s = 1, 2, ..., N. For N = w = a₁ = 1 it is the Riemann zeta function.