# Euler totient function

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 .

## 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 .

## Summatory function and average value

### Summatory function

The summatory function of the Euler phi-function is termed the totient summatory function.

## Relation with other arithmetic functions

### Similar functions

• Universal exponent (also called Carmichael function) is the exponent of the multiplicative group modulo . The universal exponent of , usually denoted , divides .
• Dedekind psi-function is similar tothe Euler phi-function, and is defined as:

.

.

.

.

• .

## 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:

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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.

## 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).