Average order of an arithmetic function: Difference between revisions

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We calculate the [[natural density]] of these numbers in {{math|'''N'''}}, that is, the average value of [[indicator function|<math>1_{Q_k}</math>]], denoted by <math>\delta(n)</math>, in terms of the [[zeta function]].

The function <math>\delta</math> is multiplicative, and since it is bounded by 1, its [[Dirichlet series]] converges absolutely in the half-plane <math>\mathrm{Re}(s)<>1</math>, and there has [[Euler product]]

: <math>\sum_{Q_k}n^{-s}=\sum_n \delta(n)n^{-s}=\prod_p (1+p^{-s}+\cdots +p^{-s(k-1)})=\prod_p \left(\frac{1-p^{-sk}}{1-p^{-s}}\right)=\frac{\zeta(s)}{\zeta(sk)}. </math>