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 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.
The proof of Euler's product formula depends on two important facts.

The function is multiplicative

This means that if, then.

Value 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 for to not equal 1 is for to be a multiple of. The multiples of that are less than or equal to are, and there are of them. 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 for,
where are prime numbers and each.
Repeatedly using the multiplicative property of and the formula for gives
This is Euler's product formula.
An alternative proof that does not require the multiplicative property may be obtained using the inclusion-exclusion principle.

Example

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 numbers that are coprime to 36. And indeed there are twelve positive integers that are coprime with 36 and lower than 36: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, and 35.

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
Unlike the other two formulae 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 way is to note that is also equal to the number of possible generators of the cyclic group ; specifically, if, then is a generator for every coprime to. Since every element of generates a cyclic subgroup, and all subgroups of are generated by some element of, the formula follows. In the article Root of unity Euler's formula is derived by using this argument in the special case of the multiplicative group of the th roots of unity.
This formula can also be derived in a more concrete manner. Let and consider the fractions between 0 and 1 with denominator 20:
Put them into lowest terms:
First note that all the divisors of 20 are denominators. And second, note that there are 20 fractions. Which fractions have 20 as denominator? The ones whose numerators are relatively prime to 20. By definition this is fractions. Similarly, there are fractions with denominator 10, fractions with denominator 5, and so on.
In detail, we are considering the fractions of the form where is an integer from 1 to inclusive. Upon reducing these to lowest terms, each fraction will have as its denominator some divisor of. We can group the fractions together by denominator, and we must show that for a given divisor of, the number of such fractions with denominator is.
Note that to reduce to lowest terms, we divide the numerator and denominator by. The reduced fractions with denominator are therefore precisely the ones originally of the form in which. The question therefore becomes: how many are there less than or equal to which verify ? Any such must clearly be a multiple of, but it must also be coprime to . Conversely, any multiple of which is coprime to will satisfy. We can generate such numbers by taking the numbers less than coprime to and multiplying each one by . This in fact generates all such numbers, as if is a multiple of coprime to , then will still be coprime to, and must also be smaller than, else would be larger than. Thus there are precisely values of less than or equal to such that, which was to be demonstrated.
Möbius inversion gives
where is the Möbius function.
This formula may also be derived from the product formula by multiplying out
to get

Some values of the function

The first 143 values are shown in the table and graph below:
The top line in the graph,, is a true upper bound. It is attained whenever is prime. There is no lower bound that is a straight line of positive slope; no matter how gentle the slope of a line is, there will eventually be points of the plot below the line. More precisely, the lower limit of the graph is proportional to rather than being linear.

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 1965 P. Kesava Menon proved
where Divisor function| 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 e |:
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. 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 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

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 are released to the public, and 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. 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. 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.