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Fungsi Phi Euler φ(m) atau ⍉(m) menyatakan kardinal himpunan bilangan asli
1
≤
n
≤
m
{\displaystyle 1\leq n\leq m}
fpb (m,n) = 1.
Leonhard Euler Swiss. w. 18 September 1783, Rusia ). Pada kisaran tahun 1750-an. Lalu, Notasi φ(m) atau ⍉(m) ditulis pertama kali oleh Gauss pada tahun
Contoh :
Bilangan bulat positif yang < 9 adalah 1, 2, 3, 4, 5, 6, 7, 8. Diantara bilangan-bilangan tersebut yang saling prima terhadap 9 adalah 1, 2, 4, 5, 7, 8, maka banyaknya bilangan yang saling prima terhadap 9 adalah sebanyak 6 sehingga φ(9) = 6.
Identitas :
φ(1) = 0
φ(2) = 1
φ(P) = P - 1 untuk P prima
φ(mn) = φ(m)φ(n) jika fpb(m,n)=1
φ(Pⁿ) = Pⁿ⁻¹ (P-1)
φ(P₁×P₂×...×Pₙ) = (P₁-1)(P₂-1)(P₃-1)...(Pₙ-1)
a
∣
b
⟹
φ
(
a
)
∣
φ
(
b
)
{\displaystyle a\mid b\implies \varphi (a)\mid \varphi (b)}
n
∣
φ
(
a
n
−
1
)
untuk setiap
a
,
n
>
1
{\displaystyle n\mid \varphi (a^{n}-1)\quad {\text{untuk setiap }}a,n>1}
φ(m,n) = Φ(m).φ(n) . Note the special cases
φ
(
2
m
)
=
{
2
φ
(
m
)
if
m
is even
φ
(
m
)
if
m
is odd
{\displaystyle \varphi (2m)={\begin{cases}2\varphi (m)&{\text{ if }}m{\text{ is even}}\\\varphi (m)&{\text{ if }}m{\text{ is odd}}\end{cases}}}
φ
(
n
m
)
=
n
m
−
1
φ
(
n
)
{\displaystyle \varphi \left(n^{m}\right)=n^{m-1}\varphi (n)}
φ
(
lcm
(
m
,
n
)
)
⋅
φ
(
gcd
(
m
,
n
)
)
=
φ
(
m
)
⋅
φ
(
n
)
{\displaystyle \varphi (\operatorname {lcm} (m,n))\cdot \varphi (\operatorname {gcd} (m,n))=\varphi (m)\cdot \varphi (n)}
Compare this to the formula
lcm
(
m
,
n
)
⋅
gcd
(
m
,
n
)
=
m
⋅
n
{\displaystyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n}
(See least common multiple .) φ (n )n ≥ 3n has r distinct odd prime factors, 2r φ (n ) For any a > 1n > 64 ∤ n there exists an l ≥ 2n l | φ (an − 1)
φ
(
n
)
n
=
φ
(
rad
(
n
)
)
rad
(
n
)
{\displaystyle {\frac {\varphi (n)}{n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}}
where rad(n ) is the radical of n .
∑
d
∣
n
μ
2
(
d
)
φ
(
d
)
=
n
φ
(
n
)
{\displaystyle \sum _{d\mid n}{\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}}
[ 1]
∑
1
≤
k
≤
n
(
k
,
n
)
=
1
k
=
1
2
n
φ
(
n
)
for
n
>
1
{\displaystyle \sum _{1\leq k\leq n \atop (k,n)=1}\!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1}
∑
k
=
1
n
φ
(
k
)
=
1
2
(
1
+
∑
k
=
1
n
μ
(
k
)
⌊
n
k
⌋
2
)
=
3
π
2
n
2
+
O
(
n
(
log
n
)
2
3
(
log
log
n
)
4
3
)
{\displaystyle \sum _{k=1}^{n}\varphi (k)={\tfrac {1}{2}}\left(1+\sum _{k=1}^{n}\mu (k)\left\lfloor {\frac {n}{k}}\right\rfloor ^{2}\right)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}
[ 2] [ 3]
∑
k
=
1
n
φ
(
k
)
k
=
∑
k
=
1
n
μ
(
k
)
k
⌊
n
k
⌋
=
6
π
2
n
+
O
(
(
log
n
)
2
3
(
log
log
n
)
4
3
)
{\displaystyle \sum _{k=1}^{n}{\frac {\varphi (k)}{k}}=\sum _{k=1}^{n}{\frac {\mu (k)}{k}}\left\lfloor {\frac {n}{k}}\right\rfloor ={\frac {6}{\pi ^{2}}}n+O\left((\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)}
[ 2]
∑
k
=
1
n
k
φ
(
k
)
=
315
ζ
(
3
)
2
π
4
n
−
log
n
2
+
O
(
(
log
n
)
2
3
)
{\displaystyle \sum _{k=1}^{n}{\frac {k}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}n-{\frac {\log n}{2}}+O\left((\log n)^{\frac {2}{3}}\right)}
[ 4]
∑
k
=
1
n
1
φ
(
k
)
=
315
ζ
(
3
)
2
π
4
(
log
n
+
γ
−
∑
p
prime
log
p
p
2
−
p
+
1
)
+
O
(
(
log
n
)
2
3
n
)
{\displaystyle \sum _{k=1}^{n}{\frac {1}{\varphi (k)}}={\frac {315\,\zeta (3)}{2\pi ^{4}}}\left(\log n+\gamma -\sum _{p{\text{ prime}}}{\frac {\log p}{p^{2}-p+1}}\right)+O\left({\frac {(\log n)^{\frac {2}{3}}}{n}}\right)}
[ 4] (where γ is the Euler–Mascheroni constant ).
∑
gcd
(
k
,
m
)
=
1
1
≤
k
≤
n
1
=
n
φ
(
m
)
m
+
O
(
2
ω
(
m
)
)
{\displaystyle \sum _{\stackrel {1\leq k\leq n}{\operatorname {gcd} (k,m)=1}}\!\!\!\!1=n{\frac {\varphi (m)}{m}}+O\left(2^{\omega (m)}\right)}
where m > 1ω (m )m .[ 5] Pengembangan Fungsi Phi Euler :
^ Dineva (in external refs), prop. 1
^ a b Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie . Mathematische Forschungsberichte (dalam bahasa German). 16 . Berlin: VEB Deutscher Verlag der Wissenschaften . Zbl 0146.06003 . ^ Lomadse, G., "The scientific work of Arnold Walfisz" (PDF) , Acta Arithmetica , 10 (3): 227–237 ^ a b Sitaramachandrarao, R. (1985). "On an error term of Landau II". Rocky Mountain J. Math . 15 : 579–588. ^ Bordellès in the external links
