Shintani zeta function
Encyclopedia
In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were studied studied by . They include Hurwitz zeta functions, Barnes zeta function
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...

s, and Witten zeta function
Witten zeta function
In mathematics, the Witten zeta function, introduced by , is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group...

s as special cases.

Definition

The Shintani zeta function of (s1, ..., sk) is given by


where each Lj is an inhomogeneous linear function of (n1, ... ,nm). The special case when k = 1 is the Barnes zeta function
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...

.
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