Average order of an arithmetic function: Difference between revisions

Revision as of 17:43, 15 July 2016

Revision as of 17:46, 15 July 2016

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: <math>Q_k :=\{n \in \mathbb{Z}\mid n \text{ is not divisible by } d^k \text{ for any integer } d\ge 2\}.</math>

We calculate the [[natural density]] of these numbers in {{math|'''N'''}}, that is, the average value of [[indicator function|<math>1_Q_k1_{Q_k}</math>]], denoted by ''δ''<math>\delta(''n'')</math>, in terms of the [[zeta function]].

The function ''δ''<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>

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</math>

Wherewhere for this we used the relation

: <math>\sum_n (f(n)/n)=\sum_n f(n^k)n^{-k}=\sum_n \mu(n)n^{-k}=\frac{1}{\zeta(k)},</math>