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==Relation to numerical methods==
Zeta function regularization gives a nice analytic structure to any sums over an [[arithmetic function]] <math>f(n)</math>. The regularized form
:<math>\tilde{f}(s) = \sum_{n=1}^\infty f(n)n^{-s}</math>
converts divergences of the sum into [[simple pole]]s on the complex ''s''-plane. In numerical calculations, the zeta function regularization is inappropriate, as it is extremely slow to converge. A far more useful sum is an exponential regularization, given by
:<math>F(t)=\sum_{n=1}^\infty f(n) e^{-tn}</math>
The analytic structure of these two functions is related.
[[Category:Quantum field theory]]
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