Barnes Zeta Function

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

: $\zeta_N(s,w\mid a_1,\ldots,a_N)=\sum_\frac$
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'' = ''a''₁ = 1 it is the Riemann zeta function.

References

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Zeta and L-functions

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