Euler totient function

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This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Let be a natural number. The Euler phi-function or Euler totient function of , denoted , is defined as following:

  • It is the order of the multiplicative group modulo , i.e., the multiplicative group of the ring of integers modulo .
  • It is the number of elements in that are relatively prime to .

In terms of prime factorization

Suppose we have the following prime factorization of :

.

Then, we have:

.

In other words:

.

Properties

Property Satisfied? Statement with symbols
multiplicative function Yes If and are relatively prime natural numbers, then .
completely multiplicative function No It is not true for arbitrary natural numbers and that . For instance, if , then whereas is 2.
divisibility-preserving function Yes If and are natural numbers such that divides , then divides .

Algebraic significance

The Euler phi-function of is important in the following ways:

  • It is the number of generators of the cyclic group of order .
  • It is the order of the multiplicative group of the ring of integers modulo (in fact, this multiplicative group is precisely the set of generators of the additive group).

Behavior

High and low points (relatively speaking)

  • Primes are high points: We also have for . Equality occurs if and only if is a prime number.
  • Primorials are low points: Roughly, the numbers occurring as primorials (products of the first few primes) have the lowest value of relative to , compared with other similarly sized numbers.

Measures of difference

We use the infinitude of primes for arguing about limit superiors. The limits discussed are in the limit as . Note that the last quotient is undefined for .

Measure Limit superior Explanation Limit inferior Explanation
-1 maximum value of -1 occurs at primes Consider the sequence of powers 2. , so .
1 At each prime , value is . Limit is 1 as 0 Consider the sequence of primorials. The corresponding values of are products of the values for the first few primes . The limit of these is the infinite product over all prime . This diverges because the infinite sum of the reciprocals of the primes diverges.
1 For each prime , the limit is ,which approaches 1. 1 We can show that for every , there exists such that for all .

Initial values

Prime factorization Totient function (equals 1 iff is prime) (reduced, equals product of for primes dividing -- depends only on the set of prime divisors) Universal exponent
1 empty 1 0 1 1 1
2 2 1 1 1/2 1 1
3 3 2 1 2/3 2 1
4 2 2 1/2 2 1
5 5 4 1 4/5 4 1
6 2 4 1/3 2 1
7 7 6 1 6/7 6 1
8 4 4 1/2 2 1
9 6 3 2/3 6 1
Prime 1 1
Power 1/2 if , else 2 if , 1 otherwise
Power , odd 1

Related arithmetic functions

Summatory functions

The sum of the values of the totient function for all natural numbers up to a given number is termed the totient summary function.

Similar functions

Name of arithmetic function Description Mathematical relation with Euler totient function
Universal exponent (also called Carmichael function) The exponent of the multiplicative group modulo . Denoted divides . This is related to the group-theoretic fact that exponent divides order.
Dedekind psi-function Defined as Structural similarity in definition. Also, for all , with equality occurring iff .

Relations expressed in terms of Dirichlet products

Below are some Dirichlet products of importance.

Statement Function Description Value of the Dirichlet product the function Description Statement in ordinary notation Proof Corollaries
the function that sends every natural number to 1 the function that sends every natural number to itself Direct combinatorial argument: each summand is the number of elements in whose gcd with with . , where is the Mobius function. This follows from the fact that .
Also, , follows from this and .
the divisor count function the divisor sum function We have and by definition. Multiply both sides of the former by and use associativity to get . Use the preceding identity to get , so

The table of relationships can be conceptualized as follows, where the cell entries are the products of their row and column headers. Note that the rows can each be deduced from one another:

(Mobius function) (Kronecker delta with 1, identity for Dirichlet product) (all ones function) (divisor count function)
no name
(identity map, sends everything to itself) no name
no name no name

Inequalities

Nature of bound (upper bound or lower) Formula Arithmetic functions used Explanation Asymptotic implication
lower is the prime-counting function, and counts the number of primes less than or equal to , while is the prime divisor count function of . Any prime less than or equal to that does not divide must be relatively prime to . , whereas . Thus, we get an asymptotic lower bound of . Note that this estimation relies on the prime number theorem.
upper is the divisor count function, counting the total number of divisors of With the exception of 1, any number in cannot be both a divisor of and relatively prime to . Nothing specifically, but it does show that for non-primes, there is a relatively sharp demarcation between and (in terms of differences, not ratios).

Relation with properties of numbers

  • Prime number: A natural number such that .
  • Polygonal number: A natural number such that is a power of , or equivalently, such that the regular -gon is constructible using straightedge and compass.


Dirichlet series

Further information: Formula for Dirichlet series of Euler phi-function

The Dirichlet series for the Euler phi-function is given by:

.

Using the Dirichlet product identity and the fact that Dirichlet series of Dirichlet product equals product of Dirichlet series, we get:

.

This simplifies to:

.

This identity holds not just for the formal Dirichlet series, but also for their analytic continuations, and is valid universally for the meromorphic functions.