Barnes zeta function explained

In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function.

Definition

The Barnes zeta function is defined by

\zeta _{N(s,w\mid} a _{1,\ldots,a} _{N)=\sum} ^{n _{1,...,n} _{N\ge} 0} (1)/((w + n _{1 a} _{1+ … +n} _{Na} ^{s} ^{N)}) 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.