Barnes zeta function
Encyclopedia
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
.
where w and aj 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function
Shintani zeta function
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were studied studied by...
.
Definition
The Barnes zeta function is defined by
where w and aj have positive real part and s has real part greater than N.
It has a meromorphic continuation to all complex s, whose only singularities
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
are simple poles at s = 1, 2, ..., N. For N = w = a1 = 1 it is the Riemann zeta function.