Average order of an arithmetic function: Difference between revisions

Revision as of 20:02, 13 August 2015

Revision as of 05:48, 14 August 2015

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: <math>q^n \text{Ave}_n\sigma_{k}=q^{n(k+1)}(\frac{1-q^{-k(n+1)}}{1-q^{-k}})=q^{n(k+1)}(\frac{\zeta(k+1)}{\zeta(kn+k+1)})</math>

Thus, if we set <math>x=q^n</math> then the above result reads

: <math>\sum_{\deg(m)=n, m \text{ monic}} \sigma_k(m)=x^{k+1}(\frac{\zeta(k+1)}{\zeta(kn+k+1)})</math>