Euler's totient function
In number theory, Euler's totient function counts the positive integers up to a given integer  that are relatively prime to . It is written using the Greek letter phi as  or , and may also be called Euler's phi function. In other words, it is the number of integers  in the range  for which the greatest common divisor  is equal to 1. The integers  of this form are sometimes referred to as totatives of .

For example, the totatives of  are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since  and . Therefore, . As another example,  since for  the only integer in the range from 1 to  is 1 itself, and .

Euler's totient function is a multiplicative function, meaning that if two numbers  and  are relatively prime, then .
This function gives the order of the multiplicative group of integers modulo  (the group of units of the ring ). It is also used for defining the RSA encryption system.

History, terminology, and notation 

Leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter  to denote it: he wrote  for "the multitude of numbers less than , and which have no common divisor with it". This definition varies from the current definition for the totient function at  but is otherwise the same. The now-standard notation  comes from Gauss's 1801 treatise Disquisitiones Arithmeticae, although Gauss didn't use parentheses around the argument and wrote . Thus, it is often called Euler's phi function or simply the phi function.

In 1879, J. J. Sylvester coined the term totient for this function, so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's.

The cototient of  is defined as . It counts the number of positive integers less than or equal to  that have at least one prime factor in common with .

Computing Euler's totient function 

There are several formulas for computing .

Euler's product formula

It states

where the product is over the distinct prime numbers dividing . (For notation, see Arithmetical function.)

An equivalent formulation for , where  are the distinct primes dividing n, is:The proof of these formulas depends on two important facts.

Phi is a multiplicative function 

This means that if , then . Proof outline: Let , ,  be the sets of positive integers which are coprime to and less than , , , respectively, so that , etc. Then there is a bijection between  and  by the Chinese remainder theorem.

Value of phi for a prime power argument 

If  is prime and , then

Proof: Since  is a prime number, the only possible values of  are , and the only way to have  is if  is a multiple of , i.e. , and there are  such multiples less than . Therefore, the other  numbers are all relatively prime to .

Proof of Euler's product formula

The fundamental theorem of arithmetic states that if  there is a unique expression  where  are prime numbers and each . (The case  corresponds to the empty product.) Repeatedly using the multiplicative property of  and the formula for  gives

This gives both versions of Euler's product formula.

An alternative proof that does not require the multiplicative property instead uses the inclusion-exclusion principle applied to the set , excluding the sets of integers divisible by the prime divisors.

Example

In words: the distinct prime factors of 20 are 2 and 5; half of the twenty integers from 1 to 20 are divisible by 2, leaving ten; a fifth of those are divisible by 5, leaving eight numbers coprime to 20; these are: 1, 3, 7, 9, 11, 13, 17, 19.

The alternative formula uses only integers:

Fourier transform

The totient is the discrete Fourier transform of the gcd, evaluated at 1. Let

where  for . Then

The real part of this formula is

For example, using  and :Unlike the Euler product and the divisor sum formula, this one does not require knowing the factors of . However, it does involve the calculation of the greatest common divisor of  and every positive integer less than , which suffices to provide the factorization anyway.

Divisor sum

The property established by Gauss, that

where the sum is over all positive divisors  of , can be proven in several ways. (See Arithmetical function for notational conventions.)

One proof is to note that  is also equal to the number of possible generators of the cyclic group  ; specifically, if  with , then  is a generator for every  coprime to . Since every element of  generates a cyclic subgroup, and all subgroups  are generated by precisely  elements of , the formula follows. Equivalently, the formula can be derived by the same argument applied to the multiplicative group of the th roots of unity and the primitive th roots of unity.

The formula can also be derived from elementary arithmetic. For example, let  and consider the positive fractions up to 1 with denominator 20:

Put them into lowest terms:

These twenty fractions are all the positive  ≤ 1 whose denominators are the divisors . The fractions with 20 as denominator are those with numerators relatively prime to 20, namely , , , , , , , ; by definition this is  fractions. Similarly, there are  fractions with denominator 10, and  fractions with denominator 5, etc. Thus the set of twenty fractions is split into subsets of size  for each  dividing 20. A similar argument applies for any n.

Möbius inversion applied to the divisor sum formula gives

where  is the Möbius function, the multiplicative function defined by  and  for each prime  and . This formula may also be derived from the product formula by multiplying out  to get 

An example:

Some values

The first 100 values  are shown in the table and graph below:

{| class="wikitable" style="text-align: right"
|+ for 
! +
! 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  is an upper bound valid for all  other than one, and attained if and only if  is a prime number. A simple lower bound is , which is rather loose: in fact, the lower limit of the graph is proportional to .

Euler's theorem

This states that if  and  are relatively prime then

The special case where  is prime is known as Fermat's little theorem.

This follows from Lagrange's theorem and the fact that  is the order of the multiplicative group of integers modulo .

The RSA cryptosystem is based on this theorem: it implies that the inverse of the function , where  is the (public) encryption exponent, is the function , where , the (private) decryption exponent, is the multiplicative inverse of  modulo . The difficulty of computing  without knowing the factorization of  is thus the difficulty of computing : this is known as the RSA problem which can be solved by factoring . The owner of the private key knows the factorization, since an RSA private key is constructed by choosing  as the product of two (randomly chosen) large primes  and . Only  is publicly disclosed, and given the difficulty to factor large numbers we have the guarantee that no one else knows the factorization.

Other formulae
Note the special cases

Compare this to the formula

(See least common multiple.)

 is even for . Moreover, if  has  distinct odd prime factors, 
 For any  and  such that  there exists an  such that .
where  is the radical of  (the product of all distinct primes dividing ).
 

 ( cited in)

 
 
 

(where  is the Euler–Mascheroni constant).

where  is a positive integer and  is the number of distinct prime factors of .

Menon's identity

In 1965 P. Kesava Menon proved

where  is the number of divisors of .

Formulae involving the golden ratio

Schneider found a pair of identities connecting the totient function, the golden ratio and the Möbius function . In this section  is the totient function, and  is the golden ratio.

They are:

and

Subtracting them gives

Applying the exponential function to both sides of the preceding identity yields an infinite product formula for :

The proof is based on the two formulae

Generating functions

The Dirichlet series for  may be written in terms of the Riemann zeta function as:

The Lambert series generating function is

which converges for .

Both of these are proved by elementary series manipulations and the formulae for .

Growth rate

In the words of Hardy & Wright, the order of  is "always 'nearly '."

First

but as n goes to infinity, for all 

These two formulae can be proved by using little more than the formulae for  and the divisor sum function .

In fact, during the proof of the second formula, the inequality

true for , is proved.

We also have

Here  is Euler's constant, , so  and .

Proving this does not quite require the prime number theorem. Since  goes to infinity, this formula shows that

In fact, more is true.

and

The second inequality was shown by Jean-Louis Nicolas. Ribenboim says "The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption."

For the average order, we have

due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov (this is currently the best known estimate of this type). The "Big " stands for a quantity that is bounded by a constant times the function of  inside the parentheses (which is small compared to ).

This result can be used to prove that the probability of two randomly chosen numbers being relatively prime is .

Ratio of consecutive values

In 1950 Somayajulu proved

In 1954 Schinzel and Sierpiński strengthened this, proving that the set

is dense in the positive real numbers. They also proved that the set

is dense in the interval (0,1).

Totient numbers
A totient number is a value of Euler's totient function: that is, an  for which there is at least one  for which . The valency or multiplicity of a totient number  is the number of solutions to this equation. A nontotient is a natural number which is not a totient number. Every odd integer exceeding 1 is trivially a nontotient. There are also infinitely many even nontotients, and indeed every positive integer has a multiple which is an even nontotient.

The number of totient numbers up to a given limit  is

for a constant .

If counted accordingly to multiplicity, the number of totient numbers up to a given limit  is

where the error term  is of order at most  for any positive .

It is known that the multiplicity of  exceeds  infinitely often for any .

Ford's theorem

 proved that for every integer  there is a totient number  of multiplicity : that is, for which the equation  has exactly  solutions; this result had previously been conjectured by Wacław Sierpiński, and it had been obtained as a consequence of Schinzel's hypothesis H. Indeed, each multiplicity that occurs, does so infinitely often.

However, no number  is known with multiplicity . Carmichael's totient function conjecture is the statement that there is no such .

Perfect totient numbers

Applications

Cyclotomy

In the last section of the Disquisitiones Gauss proves that a regular -gon can be constructed with straightedge and compass if  is a power of 2. If  is a power of an odd prime number the formula for the totient says its totient can be a power of two only if  is a first power and  is a power of 2. The primes that are one more than a power of 2 are called Fermat primes, and only five are known: 3, 5, 17, 257, and 65537. Fermat and Gauss knew of these. Nobody has been able to prove whether there are any more.

Thus, a regular -gon has a straightedge-and-compass construction if n is a product of distinct Fermat primes and any power of 2. The first few such  are
2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... .

The RSA cryptosystem

Setting up an RSA system involves choosing large prime numbers  and , computing  and , and finding two numbers  and  such that . The numbers  and  (the "encryption key") are released to the public, and  (the "decryption key") is kept private.

A message, represented by an integer , where , is encrypted by computing .

It is decrypted by computing . Euler's Theorem can be used to show that if , then .

The security of an RSA system would be compromised if the number  could be factored or if  could be computed without factoring .

Unsolved problems

Lehmer's conjecture

If  is prime, then . In 1932 D. H. Lehmer asked if there are any composite numbers  such that  divides . None are known.

In 1933 he proved that if any such  exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ). In 1980 Cohen and Hagis proved that  and that . Further, Hagis showed that if 3 divides  then  and .

Carmichael's conjecture

This states that there is no number  with the property that for all other numbers , , . See Ford's theorem above.

As stated in the main article, if there is a single counterexample to this conjecture, there must be infinitely many counterexamples, and the smallest one has at least ten billion digits in base 10.

See also 
Carmichael function
Duffin–Schaeffer conjecture
Generalizations of Fermat's little theorem
Highly composite number
Multiplicative group of integers modulo 
Ramanujan sum
Totient summatory function
Dedekind psi function

Notes

References

The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.

References to the Disquisitiones are of the form Gauss, DA, art. nnn.

. See paragraph 24.3.2.

 Dickson, Leonard Eugene, "History Of The Theory Of Numbers", vol 1, chapter 5 "Euler's Function, Generalizations; Farey Series", Chelsea Publishing 1952
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External links
 
Euler's Phi Function and the Chinese Remainder Theorem — proof that  is multiplicative
Euler's totient function calculator in JavaScript — up to 20 digits
Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
Plytage, Loomis, Polhill Summing Up The Euler Phi Function

Modular arithmetic
Multiplicative functions
Articles containing proofs
Algebra
Number theory
Leonhard Euler