Editing Euler's totient function (section)

==Some values==

The first 100 values {{OEIS|A000010}} are shown in the table and graph below:

[[File:EulerPhi100.svg|thumb|Graph of the first 100 values]]
:{| class="wikitable" style="text-align: right"
|+{{math|''φ''(''n'')}} for {{math|1 ≤ ''n'' ≤ 100}}
! +
! 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10
|- 
! 0 
| 1 || 1 || 2 || 2 || 4 || 2 || 6 || 4 || 6 || 4 
|-
! 10
| 10 || 4 || 12 || 6 || 8 || 8 || 16 || 6 || 18 || 8 
|- 
! 20
| 12 || 10 || 22 || 8 || 20 || 12 || 18 || 12 || 28 || 8 
|-
! 30
| 30 || 16 || 20 || 16 || 24 || 12 || 36 || 18 || 24 || 16 
|-
! 40
| 40 || 12 || 42 || 20 || 24 || 22 || 46 || 16 || 42 || 20
|-
! 50
| 32 || 24 || 52 || 18 || 40 || 24 || 36 || 28 || 58 || 16 
|-
! 60
| 60 || 30 || 36 || 32 || 48 || 20 || 66 || 32 || 44 || 24 
|-
! 70
| 70 || 24 || 72 || 36 || 40 || 36 || 60 || 24 || 78 || 32
|-
! 80
| 54 || 40 || 82 || 24 || 64 || 42 || 56 || 40 || 88 || 24
|-
! 90
| 72 || 44 || 60 || 46 || 72 || 32 || 96 || 42 || 60 || 40 
|}

In the graph at right the top line {{math|''y'' {{=}} ''n'' − 1}} is an [[upper bound]] valid for all {{mvar|n}} other than one, and attained if and only if {{mvar|n}} is a prime number. A simple lower bound is <math>\varphi(n) \ge \sqrt{n/2} </math>, which is rather loose: in fact, the [[Limit superior and limit inferior|lower limit]] of the graph is proportional to {{math|{{sfrac|''n''|log log ''n''}}}}.<ref name="hw328"/>
{{clear}}