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
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by . They include Hurwitz zeta functions and Barnes zeta functions.
Definition
Let P(\mathbf) be a polynom ...
.
Definition
The Barnes zeta function is defined by
:
where ''w'' and ''a''
''j'' have positive real part and ''s'' has real part greater than ''N''.
It has a
meromorphic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a n ...
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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