Zeta Function Regularization - Relation To Other Regularizations

Relation To Other Regularizations

Zeta-function regularization gives a nice analytic structure to any sums over an arithmetic function f(n). Such sums are known as Dirichlet series. The regularized form

converts divergences of the sum into simple poles on the complex s-plane. In numerical calculations, the zeta-function regularization is inappropriate, as it is extremely slow to converge. For numerical purposes, a more rapidly converging sum is the exponential regularization, given by

This is sometimes called the Z-transform of f, where z = exp(−t). The analytic structure of the exponential and zeta-regularizations are related. By expanding the exponential sum as a Laurent series

one finds that the zeta-series has the structure

The structure of the exponential and zeta-regulators are related by means of the Mellin transform. The one may be converted to the other by making use of the integral representation of the Gamma function:

which lead to the identity

relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series.