n ∏ p ∣ n ( 1 − 1 p ) , \left(1-{\frac {1}{p}}\right),} where the product is over the distinct prime numbers dividing n. An equivalent formulation is φ ( n )
n φ ( n ) {\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}} ∑ 1 ≤ k ≤ n ( k , n )
[ m ]∑ k1 n x k ⋅ e − 2 π i m k n }\{\mathbf {x} \}[m]=\sum \limits _{k=1}^{n}x_{k}\cdot e^{{-2\pi i}{\frac {mk}{n}}}} where xkgcd(k,n) for k ∈ {1, ..., n}.
1 1 ≤ k ≤ n 1n φ ( m ) m + O ( 2 ω ( m ) ) {\operatorname {gcd} (k,m)=1}}\!\!\!\!1=n{\frac {\varphi (m)}{m}}+O\left(2^{\omega (m)}\right)} dimana m > 1 adalah bilangan bulat positif dan ω ( m ) adalah jumlah faktor prima yang berbeda dari m .
}\{\mathbf {x} \}[1]=\sum \limits _{k=1}^{n}\gcd(k,n)e^{-2\pi i{\frac {k}{n}}}.}
n φ ( n ) {\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}} ∑ 1 ≤ k ≤ n ( k , n )
∑ k1 n gcd ( k , n )
∑ gcd ⁡ ( k , m )
m ∣ φ ( a m − 1 ) -1)} φ ( m n )
φ ( m ) ⋅ φ ( n ) Perhatikan kasus khusus φ ( 2 m ){ 2 φ ( m )
m ⋅ n {\textstyle \operatorname {lcm} (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n} (see least common multiple). φ(n) is even for n ≥ 3.
m ⋅ n (m,n)\cdot \operatorname {gcd} (m,n)=m\cdot n} (Lihat kelipatan persekutuan terkecil.)
φ ( n ) nφ ( rad ⁡ ( n ) ) rad ⁡ ( n ) {n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}} where rad(n) is the radical of n (the product of all distinct primes dividing n).
φ ( n ) nφ ( rad ⁡ ( n ) ) rad ⁡ ( n ) {n}}={\frac {\varphi (\operatorname {rad} (n))}{\operatorname {rad} (n)}}} di mana rad ⁡ ( n ) (n)} adalah radikal dari n .
n φ ( n ) {\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}} ∑ 1 ≤ k ≤ n − 1 g c d ( k , n )
n φ ( n ) {\frac {\mu ^{2}(d)}{\varphi (d)}}={\frac {n}{\varphi (n)}}} ∑ 1 ≤ k ≤ n ( k , n )
1 k1 2 n φ ( n ) for n > 1 \!\!k={\tfrac {1}{2}}n\varphi (n)\quad {\text{for }}n>1} ∑ k1 n φ ( k )1 2 ( 1 + ∑ k1 n μ ( k )
1 k1 2 n φ ( n ) \!\!k={\tfrac {1}{2}}n\varphi (n)} , untuk n > 1 ∑ k1 n φ ( k )1 2 ( 1 + ∑ k1 n μ ( k )
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ( cited in) ∑ k1 n φ ( k )
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ( dikutip dalam) ∑ k1 n φ ( k )
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 1 3 ) ^{n}\varphi (k)={\frac {3}{\pi ^{2}}}n^{2}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {1}{3}}\right)}
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ( dikutip dalam) ∑ k1 n φ ( k )
[Liu (2016)] ∑ k1 n φ ( k ) k∑ k1 n μ ( k ) k ⌊ n k ⌋6 π 2 n + O ( ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ∑ k1 n k φ ( k )315 ζ ( 3 ) 2 π 4 n − log ⁡ n 2 + O ( ( log ⁡ n ) 2 3 ) ^{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)} ∑ k1 n 1 φ ( k )
k∑ k1 n μ ( k ) k ⌊ n k ⌋6 π 2 n + O ( ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ∑ k1 n k φ ( k )315 ζ ( 3 ) 2 π 4 n − log ⁡ n 2 + O ( ( log ⁡ n ) 2 3 ) ^{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)} ∑ k1 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 ) ^{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)} (dengan γ adalah konstanta Euler–Mascheroni).
315 ζ ( 3 ) 2 π 4 ( log ⁡ n + γ − ∑ p prime log ⁡ p p 2 − p + 1 ) + O ( ( log ⁡ n ) 2 3 n ) ^{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)} (where γ is the Euler–Mascheroni constant).
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ( dikutip dalam) ∑ k1 n φ ( k )
The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as: ∑ n1 ∞ φ ( n ) n sζ ( s − 1 ) ζ ( s ) ^{\infty }{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}}} where the left-hand side converges for ℜ ( s ) > 2 .
Deret Dirichlet untuk φ ( n ) dapat ditulis dalam istilah fungsi zeta Riemann sebagai: ∑ n1 ∞ φ ( n ) n sζ ( s − 1 ) ζ ( s ) .
3 n 2 π 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) as n → ∞ , }{\pi ^{2}}}+O\left(n(\log n)^{\frac {2}{3}}(\log \log n)^{\frac {4}{3}}\right)\quad {\text{as }}n\rightarrow \infty ,} due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov.
3 π 2 n 2 + O ( n ( log ⁡ n ) 2 3 ( log ⁡ log ⁡ n ) 4 3 ) ^{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)} ( dikutip dalam) ∑ k1 n φ ( k )
They also proved that the set { φ ( n ) n , n1 , 2 , … } {n}},\;\;n=1,2,\ldots \right\}} is dense in the interval (0,1).
Mereka pun membuktikannya bahwa himpunan { φ ( n ) n , n1 , 2 , … } {n}},\;\;n=1,2,\ldots \right\}} padat dalam interval ( 0 , 1 ) .