to other regularizations Zeta function regularization is equivalent to dimensional regularization, see. However, the main advantage of the zeta regularization is that it can be used whenever the dimensional regularization fails, for example if there are matrices or tensors inside the calculations Relation to Dirichlet series Zeta-function regularization gives an 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 leads to the identity relating the exponential and zeta-regulators, and converting poles in the s-plane to divergent terms in the Laurent series. Heat kernel regularization The sum is sometimes called a heat kernel or a heat-kernel regularized sum; this name stems from the idea that the can sometimes be understood as eigenvalues of the heat kernel. In mathematics, such a sum is known as a generalized Dirichlet series; its use for averaging is known as an Abelian mean. It is closely related to the Laplace–Stieltjes transform, in that where is a step function, with steps of at . A number of theorems for the convergence of such a series exist. For example, by the Hardy-Littlewood Tauberian theorem, if then the series for converges in the half-plane and is uniformly convergent on every compact subset of the half-plane . In almost all applications to physics, one has History Much of the early work establishing the convergence and equivalence of series regularized with the heat kernel and zeta function regularization methods was done by G. H. Hardy and J. E. Littlewood in 1916 and is based on the application of the Cahen–Mellin integral. The effort was made in order to obtain values for various ill-defined, conditionally convergent sums appearing in number theory. In terms of application as