The Laurent expansion of [[zeta].sub.K](s) at s = 1 is
In case K = Q, the Laurent expansion of the Riemann zeta function [zeta](s) at its pole s = 1 is given by
Israilov, The Laurent expansion of the Riemann zeta function, Trudy Mat.
If c(q) [not equal to] 0, then it has a double pole at the point s = 1, and the main part of its Laurent expansion
at this point is
In the last ten years, much work has been done on Laurent expansion formulas for cluster algebras.
In this paper, we study the Newton polytope of the Laurent expansion of a type A cluster variable with respect to an arbitrary cluster.
Linner  simplified Brugia's method by using operators and extended it to improper functions by Laurent expansion. In , a modification was also made to Brugia's method, which leads to relative simpler implementation and a review of many nice "aged" methods can also be found in .
Laurent Expansion Applied to Improper Rational Functions for Any sc.
Proof." The function [S.sub.j](t) can be obtained from the Laurent expansion of [a.sub.j](z) in an annulus containing the unit circle, i.e., [a.sub.j](z) := [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII]
, s, the coefficients [c.sub.j,n] in the Laurent expansion [a.sub.j](z) [MATHEMATICAL EXPRESSIONS NOT REPRODUCIBLE IN ASCII] in the annulus [C.sub.[rho]] satisfy
To be more specific, the formulas for the constants [k.sub.s] and [k.sub.s-1] that we find are sums involving coefficients in the Laurent expansion of certain Dirichlet series.
as n [right arrow [infinity] where [C.sub.2a] is the constant term in the Laurent expansion of [Z.sub.2a] (z) around its pole and [Statment] is defined by 1.