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 . It is also used for defining the RSA encryption system.

History, terminology, and notation

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 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, 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 .
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. 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.
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. 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:
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 encryption exponent, is the function, where, the 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 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:
  • is even for.
Moreover, if has distinct odd prime factors,
.

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. 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.
By a combination of van der Corput's and Vinogradov's methods, H.-Q. Liu
improved the error term to
. The "Big " stands for a quantity that is bounded by a constant times the function of inside the parentheses.
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.

Totient number

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 first few totient numbers are, see sequence .
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.