Average order of an arithmetic function: Difference between revisions

Revision as of 06:22, 10 August 2014

Revision as of 06:23, 10 August 2014

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For example, the product representation of the zeta function is as for the integers: <math>\zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}</math>.

Unlike the classical [[zeta function]], <math>\zeta_{A}(s)</math> is verya simple rational function:

<math>\zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\text{deg(f)=n}}q^{-sn}=\sum_{n}(q^{n-sn})=(1-q^{1-s})^{-1}</math>.