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It is conventional to choose an approximating function ''g'' that is [[Continuous function|continuous]] and [[Monotonic function|monotone]]. But even thus an average order is of course not unique.
==Calculating mean values using Dirichlet series==
In case ''F'' is of the form
<math>F(n)=\sum_{d \mathop |n} f(n),</math></br> for some arithmetic function ''f(n)'', one has,
<math>\sum_{n \le x} F(n)=\sum_{d \le x} f(d) \sum_{n\le x, d|x} 1=\sum_{f \le d} f(d)[x/d] = x\sum_{d \le x} \frac{f(d)}{d} \text{ } + O(\sum_{d \le x} |f(d)|).\qquad\qquad (1)</math>
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 power free integers in {{math|'''N'''}}===
For an integer ''k''≥1 the set ''Q<sub>k</sub>'' of '''''k''-th-power-free''' integers is
<math>Q_{k}:=\{n \in \mathbb{Z}|\;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|'''1<sub>Q<sub>k</sub></sub>''']], denoted by ''δ(n)'', in terms of the [[zeta function]].
The function ''δ'' is multiplicative, and since it is bounded by 1, its [[Dirichlet series]] converges absolutely in the half-plane Re(s)>1, 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>.
By the [[Möbius inversion]] formula, we get
<math>
\frac{1}{\zeta(ks)}=\sum_{n}\mu(n)n^{-ks},
</math>
where <math>\mu</math> stands for the [[Möbius function]]. Equivalently,
<math>
\frac{1}{\zeta(ks)}=\sum_{n}f(n)n^{-s},
</math>
where <math>f(n)=\begin{cases}
\;\;\, \mu(d) & n=d^{k}\\
\;\;\, 0 & \text{otherwise},
\end{cases}</math>
and hence,
<math>\frac{\zeta(s)}{\zeta(sk)}=\sum_{n}(\sum_{d|n}f(d))n^{-s}</math>.
By comparing the coefficients, we get
<math>\delta(n)=\sum_{d|n}f(d)n^{-s}</math>.
Using (1), we get
<math>\sum_{d \le x}\delta(d)=x\sum_{d \le x}(f(d)/d)+O(x^{1/k})</math>.
We conclude that,
<math>
\sum_{n\in Q_{k}, n \le x}1=\frac{x}{\zeta(k)}+O(x^{1/k})
</math>,
Where 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>,
which follows from the Möbius inversion formula.
In particular, the density of the [[square-free integers]] is <math>\zeta(2)^{-1}=\frac{6}{\pi^{2}}</math>.
==Examples==
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