Average order of an arithmetic function: Difference between revisions

Revision as of 06:31, 10 August 2014

Revision as of 06:33, 10 August 2014

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<math> \text{Ave}_{n}\sigma_{k}=\frac{1-q^{k(n+1)}}{1-q^{k}}</math>.

'''====Number of divisors'''====

Let <math>d(f)</math> be the number of monic divisors of ''f'' and let <math>D(n)</math> be the sum of <math>d(f)</math> over all monics of degree n.

<math>\zeta_{A}(s)^{2}=(\sum_{h}|h|^{-s})(\sum_{g}|g|^{-s})=\sum_{f}(\sum_{hg=f}1)|f|^{-s}=\sum_{f}d(f)}|f|^{-s}=D_{d}(s)=\sum_{n \mathop =0}^{\infty}D(n)u^{n}</math>

where <math>u=q^{-s}</math>.