Average order of an arithmetic function: Difference between revisions

Revision as of 17:39, 15 July 2016

Revision as of 17:43, 15 July 2016

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==Calculating mean values using Dirichlet series==

In case ''<math>F''</math> is of the form

: <math>F(n)=\sum_{d \mid n} f(d),</math>

for some arithmetic function ''<math>f''(''n'')</math>, one has,

: <math>\sum_{n \le x} F(n)=\sum_{d \le x} f(d) \sum_{n\le x, d\mid n} 1=\sum_{d \le x} 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>

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

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

: <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|'''1<sub>''Q''<sub>''k''</submath>1_Q_k</submath>''']], 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]]