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 ).See Euler's theorem. It is also used for defining the RSA encryption system.
History, terminology, and notation
Leonhard Euler introduced the function in 1763.L. Euler "Theoremata arithmetica nova methodo demonstrata" (An arithmetic theorem proved by a new method), Novi commentarii academiae scientiarum imperialis Petropolitanae (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), 8 (1763), 74–104. (The work was presented at the Saint-Petersburg Academy on October 15, 1759. A work with the same title was presented at the Berlin Academy on June 8, 1758). Available on-line in: Ferdinand Rudio, , Leonhardi Euleri Commentationes Arithmeticae, volume 1, in: Leonhardi Euleri Opera Omnia, series 1, volume 2 (Leipzig, Germany, B. G. Teubner, 1915), pages 531–555. On page 531, Euler defines as the number of integers that are smaller than and relatively prime to (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N).Sandifer, p. 203Graham et al. p. 133 note 111 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".L. Euler, Speculationes circa quasdam insignes proprietates numerorum, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115. (The work was presented at the Saint-Petersburg Academy on October 9, 1775). This definition varies from the current definition for the totient function at but is otherwise the same. The now-standard notationBoth and are seen in the literature. These are two forms of the lower-case Greek letter phi. comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,Gauss, Disquisitiones Arithmeticae article 38 although Gauss did not 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,J. J. Sylvester (1879) "On certain ternary cubic-form equations", American Journal of Mathematics, 2 : 357-393; Sylvester coins the term "totient" on page 361. 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 formulae for computing .
Euler's product formula
It states
where the product is over the distinct prime numbers dividing .
An equivalent formulation is
where is the prime factorization of (that is, are distinct prime numbers).
The proof of these formulae 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 , that is, , and there are such multiples not greater 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,Gauss, DA, art 39 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 each subgroup is generated by precisely elements of , the formula follows.Gauss, DA art. 39, arts. 52-54 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.Graham et al. pp. 134-135 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
In particular:
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 ). Dineva (in external refs), prop. 1
( cited in) [Liu (2016)]
(where is the Euler–Mascheroni constant).
Menon's identity
In 1965 P. Kesava Menon proved
where is the number of divisors of .
Divisibility by any fixed positive integer
The following property, which is unpublished as a specific result but has long been known, has important consequences. For instance it rules out uniform distribution of the values of in the arithmetic progressions modulo for any integer .
For every fixed positive integer , the relation holds for almost all , meaning for all but values of as .
This is an elementary consequence of the fact that the sum of the reciprocals of the primes congruent to 1 modulo diverges, which itself is a corollary of the proof of Dirichlet's theorem on arithmetic progressions.
Generating functions
The Dirichlet series for may be written in terms of the Riemann zeta function as:
where the left-hand side converges for .
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.In fact Chebyshev's theorem () and Mertens' third theorem is all that is needed. Since goes to infinity, this formula shows that
In fact, more is true.Theorem 15 of Bach & Shallit, thm. 8.8.7
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 haveSándor, Mitrinović & Crstici (2006) pp.24–25
due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov. By a combination of van der Corput's and Vinogradov's methods, H.-Q. Liu (On Euler's function.Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 4, 769–775) improved the error term to