Euler's totient function

The first thousand values of $\varphi(n)$

In number theory, the totient $\varphi(n)$ of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n (i.e. having no common positive factors other than 1). In particular $\varphi(1)=1$ since 1 is coprime to itself (1 being the only natural number with this property). For example, $\varphi(9)=6$ since the six numbers 1, 2, 4, 5, 7 and 8 are coprime to 9. The function $\varphi$ so defined is the totient function. The totient is usually called the Euler totient or Euler's totient, after the Swiss mathematician Leonhard Euler, who studied it. The totient function is also called Euler's phi function or simply the phi function, since it is commonly denoted by the Greek letter phi ( $\varphi$ ). The cototient of n is defined as $n - \varphi(n)$ , in other words the number of positive integers less than or equal to n that are not coprime to n.

The totient function is important mainly because it gives the size of the multiplicative group of integers modulo n. More precisely, $\varphi(n)$ is the order of the group of units of the ring $\mathbb{Z}/n\mathbb{Z}$ . This fact, together with Lagrange's theorem on the possible sizes of subgroups of a group, provides a proof for Euler's theorem that $a^{\varphi(n)}\equiv 1 \pmod{n}$ for all a coprime to n. The totient function also plays a key role in the definition of the RSA encryption system.

1 Computing Euler's function
- 1.1 Computing example
2 Some values of the function
3 Properties
4 Generating functions
5 Growth of the function
6 Other formulas involving Euler's function
7 Inequalities
8 Ford's theorem
9 See also
10 References
11 External links

Computing Euler's function

If p is prime, then for integer $k\ge1, \varphi(p^{k}) = (p - 1)p^{k - 1}$ . Moreover, $\varphi$ is a multiplicative function; if m and n are coprime then $\varphi(mn) = \varphi(m) \varphi(n)$ . (Sketch of proof: let A, B, C be the sets of residue classes modulo-and-coprime-to m, n, mn respectively; then there is a bijection between A × B and C, by the Chinese remainder theorem.) The value of $\varphi(n)$ can thus be computed using the fundamental theorem of arithmetic: if

$n = p_1^{k_1} \cdots p_r^{k_r}$

where each p_i is a distinct prime, then

$\varphi(n)=(p_1-1)p_1^{k_1-1} \times (p_2-1)p_2^{k_2-1} \cdots \times(p_r-1)p_r^{k_r-1}.$

This last formula is an Euler product and is often written in the equivalent form

$\varphi(n)= n \cdot \prod_{p_i|n} \left( 1-\frac{1}{p_i} \right)$

with the product ranging only over the distinct primes p dividing n.

Computing example

$\varphi(36)=\varphi\left(2^2 3^2\right)=36\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)=36\cdot\frac{1}{2}\cdot\frac{2}{3}=12.$

In words, this says that the distinct prime factors of 36 are 2 and 3; half of the thirty-six integers from 1 to 36 are divisible by 2, leaving eighteen; a third of those are divisible by 3, leaving twelve coprime to 36. And indeed there are twelve: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.

Moreover Schramm (2008) has shown that:

$\varphi (n)=\sum\limits_{k=1}^n{\operatorname{gcd}(k,n)\cdot \exp \left(\frac{-2\pi ik}{n}\right)}.$

Some values of the function

The first 99 values (sequence A000010 in OEIS) are shown in the table and graph below:

Run chart of the first 100 values

$\varphi(n)$	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9
0+		1	1	2	2	4	2	6	4	6
10+	4	10	4	12	6	8	8	16	6	18
20+	8	12	10	22	8	20	12	18	12	28
30+	8	30	16	20	16	24	12	36	18	24
40+	16	40	12	42	20	24	22	46	16	42
50+	20	32	24	52	18	40	24	36	28	58
60+	16	60	30	36	32	48	20	66	32	44
70+	24	70	24	72	36	40	36	60	24	78
80+	32	54	40	82	24	64	42	56	40	88
90+	24	72	44	60	46	72	32	96	42	60

Note that the lower bound of y = 4n/15 ≈ .267 n only occurs where n is a multiple of 30. (This is not a lower bound for the whole totient function, but only for these first few values of n. For example, $\varphi(210) = 48 = 8(210)/35 \approx .229 n$ ; $\varphi(2310) = 480 = 16(2310)/77 \approx .208 n$ ; and $\varphi(30030) = 5760 = 192(30030)/1001 \approx .192 n$ .)

Properties

$\varphi(p^\alpha) = p^\alpha - p^{\alpha - 1}$ for prime p and $\alpha \geq 1$ .
$\varphi(mn) = \varphi(m)\varphi(n)(d/\varphi(d))$ where d = gcd(m,n).
$a\mid b$ implies $\varphi(a)\mid\varphi(b)$ .
$\varphi(n)$ is even for $n \geq 3$ . Moreover, if n has r distinct odd prime factors, $2^r \mid \varphi(n)$ .

The number $\varphi(n)$ is also equal to the number of possible generators of the cyclic group C_n (and therefore also to the degree of the cyclotomic polynomial $\varphi_n$ ). Since every element of C_n generates a cyclic subgroup and the subgroups of C_n are of the form C_d where d divides n (written as d | n), we get

$\sum_{d\mid n}\varphi(d)=n$

where the sum extends over all positive divisors d of n.

We can now use the Möbius inversion formula to "invert" this sum and get another formula for $\varphi(n)$ :

$\varphi(n)=\sum_{d\mid n} d \cdot \mu\left(\frac{n}{d} \right)$

where μ is the usual Möbius function defined on the positive integers.

According to Euler's theorem, if a is coprime to n, that is, gcd(a, n) = 1, then

$a^{\varphi(n)} \equiv 1\mod n.\,$

This follows from Lagrange's theorem and the fact that a belongs to the multiplicative group of $\mathbb{Z}/n\mathbb{Z}$ iff a is coprime to n.

Generating functions

The two generating functions presented here are both consequences of the fact that

$\sum_{d|n} \varphi(d) = n.$

A Dirichlet series involving $\varphi$ (n) is

$\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}$

where ζ(s) is the Riemann zeta function. This is derived as follows:

$\zeta(s) \sum_{f=1}^\infty \frac{\varphi(f)}{f^s} = \left(\sum_{g=1}^\infty \frac{1}{g^s}\right)\left(\sum_{f=1}^\infty \frac{\varphi(f)}{f^s}\right)$

$\left(\sum_{g=1}^\infty \frac{1}{g^s}\right) \left(\sum_{f=1}^\infty \frac{\varphi(f)}{f^s}\right) = \sum_{h=1}^\infty \left(\sum_{fg=h} 1 \cdot \varphi(g)\right) \frac{1}{h^s}$

$\sum_{h=1}^\infty \left(\sum_{fg=h} \varphi(g) \right) \frac{1}{h^s} = \sum_{h=1}^\infty \left(\sum_{d|h} \varphi(d) \right) \frac{1}{h^s}$

$\sum_{h=1}^\infty \left(\sum_{d|h} \varphi(d) \right) \frac{1}{h^s} =$ $\sum_{h=1}^\infty\frac{h}{h^s}$

$\sum_{h=1}^\infty \frac{h}{h^s} = \zeta(s-1).$

A Lambert series generating function is

$\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n}= \frac{q}{(1-q)^2}$

which converges for |q|<1.

This follows from

$\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n} = \sum_{n=1}^{\infty} \varphi(n) \sum_{r\ge 1} q^{rn}$

which is

$\sum_{k\ge 1} q^k \sum_{n|k} \varphi(n) = \sum_{k\ge 1} k q^k = \frac{q}{(1-q)^2}.$

Growth of the function

The growth of $\varphi(n)$ as a function of n is an interesting question, since the first impression from small n that $\varphi(n)$ might be noticeably smaller than n is somewhat misleading. Asymptotically we have

$\,n^{1-\varepsilon}<\varphi(n)<n$

for any given ε > 0 and n > N(ε). In fact if we consider

$\,\varphi(n)/n,$

we can write that, from the formula above, as the product of factors

$1-p^{-1}\,$

taken over the prime numbers p dividing n. Therefore the values of n corresponding to particularly small values of the ratio are those n that are the product of an initial segment of the sequence of all primes. From the prime number theorem it can be shown that a constant ε in the formula above can therefore be replaced by

$C\,\log \log n/ \log n.$

$\varphi$ is also generally close to n in an average sense:

$\frac{1}{n^2} \sum_{k=1}^n \varphi(k)= \frac{3}{\pi^2} + \mathcal{O}\left(\frac{\log n }{n}\right)$

where the big O is the Landau symbol. This also says that the probability of two positive integers chosen at random from $\{1, 2, \ldots , n \}$ being relatively prime approaches $\frac{6}{\pi^2}$ when n tends to infinity. A related result is the average order of $\frac{\varphi(n)}{n}$ , which is described by

$\frac{1}{n} \sum_{k=1}^n \frac{\varphi(k)}{k} = \frac{6}{\pi^2} + \mathcal{O}\left(\frac{\log n }{n}\right)$

Because $\frac{6}{\pi^2} = \frac{1}{\zeta(2)}$ , one can also express the formula this way.

$\frac{1}{n} \sum_{k=1}^n \frac{\varphi(k)}{k} = \frac{1}{\zeta(2)} + \mathcal{O}\left(\frac{\log n }{n}\right)$

A proof of these formulas may be found here.

Other formulas involving Euler's function

$\;\varphi\left(n^m\right) = n^{m-1}\varphi(n) \text{ for } m\ge 1$

$\text{For any } a, n > 1 \text{, } n \mid \varphi(a^n-1)$

For any $a > 1$ and $n > 6$ such that $4 \nmid n$ there exists an $l \geq 2n$ such that $l \mid \varphi(a^n-1)$ .

$\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)} = \frac{n}{\varphi(n)}$

$\sum_{1\le k\le n \atop (k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)\text{ for }n>1$ See proof here.

$\sum_{k=1}^n\varphi(k) = \frac{1}{2}\left(1+ \sum_{k=1}^n \mu(k)\left\lfloor\frac{n}{k}\right\rfloor^2\right)$

$\sum_{k=1}^n\frac{\varphi(k)}{k} = \sum_{k=1}^n\frac{\mu(k)}{k}\left\lfloor\frac{n}{k}\right\rfloor$

$\sum_{k=1}^n\frac{k}{\varphi(k)} = \mathcal{O}(n)$

$\sum_{k=1}^n\frac{1}{\varphi(k)} = \mathcal{O}(\log(n))$

$\sum_{1\le k\le n \atop (k,m)=1} 1 = n \frac {\varphi(m)}{m} + \mathcal{O} \left ( 2^{\omega(m)} \right ),$

where m > 1 is a positive integer and ω(m) designates the number of distinct prime factors of m. (This formula counts the number of naturals less than or equal to n and relatively prime to m, additional material is listed among the external links.)

Proofs of some of these identities may be found here.

Inequalities

Some inequalities involving the $\varphi$ function are:

$\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}}$ for n > 2, where γ is Euler's constant,

$\varphi(n) \ge \sqrt{\frac {n} {2} }$ for n > 0,

and

$\varphi(n) \ge \sqrt{n}\text{ for } n > 6.$

For prime n, clearly $\varphi(n) = n-1$ .

For a composite number n we have

$\varphi(n) \le n-\sqrt{n}$ .

For any even number 2n where 2n is not of the form 2^k we have

$\varphi(2n) \le n-1$ .

For any even number 2n where 2n is of the form 2^k we have

$\varphi(2n) = n$ .

For arbitrarily large n, these bounds still cannot be improved, or to be more precise:

$\liminf \frac{\varphi (n)}{n}=0 \mbox{ and } \limsup \frac{\varphi (n)}{n}=1.$

A pair of inequalities combining the $\varphi$ function and the $\sigma$ divisor function are:

$\frac {6 n^2}{\pi^2} < \varphi(n) \sigma(n) < n^2 \mbox{ for } n>1.$

The last two are proved on the page on proofs of totient identities.

Ford's theorem

Ford (1999) proved that for every integer k ≥ 2 there is a number m for which the equation φ(x) = m has exactly k solutions; this result had previously been conjectured by Wacław Sierpiński. However, no such m is known for k = 1, and according to Carmichael's totient function conjecture it is believed that in this case no such m exists.

References

Abramowitz, M.; Stegun, I. A. (1964), Handbook of Mathematical Functions, New York: Dover Publications, ISBN 0-486-61272-4 . See paragraph 24.3.2.

Bach, E.; Shallit, J. (1996), Algorithmic Number Theory, 1, Cambridge, MA: MIT Press, ISBN 0-262-02405-5 . See page 234 in section 8.8.

Ford, K. (1999), "The number of solutions of φ(x) = m", Annals of Mathematics 150 (1): 283–311, doi:10.2307/121103, MR 1715326 .

Schramm, Wolfgang (2008), "The Fourier transform of functions of the greatest common divisor", Electronic Journal of Combinatorial Number Theory A50 (8(1)), http://www.integers-ejcnt.org/vol8.html .

External links

Kirby Urner, Computing totient function in Python and scheme, (2003)
Simple java script totient calculator , [1]
Miyata, Daisuke & Yamashita, Michinori, Derived logarithmic function of Euler's function
Bordellès, Olivier, Numbers prime to q in $[1, n]$
Calculate φ(n) for a number up to 2³¹ [2]
Plytage, Loomis, Polhill Summing Up The Euler Phi Function
Fabrizio Romano, a faster python implementation [3]

Totient function

Euler's totient function $\phi(n)$ · Jordan's totient function $J_k(n)$ · Nontotient · Noncototient · Highly totient number · Sparsely totient number · Highly cototient number

$\varphi(n)$	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9
0+		1	1	2	2	4	2	6	4	6
10+	4	10	4	12	6	8	8	16	6	18
20+	8	12	10	22	8	20	12	18	12	28
30+	8	30	16	20	16	24	12	36	18	24
40+	16	40	12	42	20	24	22	46	16	42
50+	20	32	24	52	18	40	24	36	28	58
60+	16	60	30	36	32	48	20	66	32	44
70+	24	70	24	72	36	40	36	60	24	78
80+	32	54	40	82	24	64	42	56	40	88
90+	24	72	44	60	46	72	32	96	42	60

$\varphi(n)$	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9
0+		1	1	2	2	4	2	6	4	6
10+	4	10	4	12	6	8	8	16	6	18
20+	8	12	10	22	8	20	12	18	12	28
30+	8	30	16	20	16	24	12	36	18	24
40+	16	40	12	42	20	24	22	46	16	42
50+	20	32	24	52	18	40	24	36	28	58
60+	16	60	30	36	32	48	20	66	32	44
70+	24	70	24	72	36	40	36	60	24	78
80+	32	54	40	82	24	64	42	56	40	88
90+	24	72	44	60	46	72	32	96	42	60