Average order of an arithmetic function: Difference between revisions

Revision as of 05:15, 23 March 2024

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Generalized identities of the previous form are found [[Divisor_sum_identities#Average_order_sum_identities|here]]. This identity often provides a practical way to calculate the mean value in terms of the [[Riemann zeta function]]. This is illustrated in the following example.

===The density of the ''k''^th-th power -free integers in {{mathmathbb|N}}===

For an integer <math>k \geq 1</math> the set <math>Q_k</math> of '''''k''-th-power-free''' integers is

<math display="block">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 {{mathmathbb|'''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>\operatorname{Re}(s)>1</math>, and there has [[Euler product]]

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<math display="block">\lim_{N\to\infty} \frac{1}{N}\sum_{n\le N} \frac{\varphi(n)}{n} = \frac{6}{\pi^2}=\frac{1}{\zeta(2)}. </math>

<math display="inline">\frac{1}{\zeta(2)}</math> is also the natural density of the square-free numbers in {{mathmathbb|'''N'''}}. In fact, this is not a coincidence. Consider the ''k''-dimensional lattice, <math>\mathbb{Z}^{k}</math>. The natural density of the points which are visible from the origin is <math display="inline">\frac{1}{\zeta(k)}</math>, which is also the natural density of the ''k''-th free integers in {{mathmathbb|'''N'''}}.

===Divisor functions===