In

Euler's Phi Function and the Chinese Remainder Theorem — proof that is multiplicative

*Dineva, Rosica

The Euler Totient, the Möbius, and the Divisor Functions

*Plytage, Loomis, Polhil

Summing Up The Euler Phi Function

{{Totient Modular arithmetic Multiplicative functions Articles containing proofs Algebra Number theory Leonhard Euler

number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, Euler's totient function counts the positive integers up to a given integer that are relatively prime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

to . It is written using the Greek letter phi
Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ; Modern Greek: ''fi'' ) is the 21st letter of the Greek alphabet.
In Archaic Greek, Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an Aspir ...

as $\backslash varphi(n)$ or $\backslash phi(n)$, 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
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

is equal to 1. The integers of this form are sometimes referred to as totative In number theory, a totative of a given positive integer is an integer such that and is coprime to . Euler's totient function φ(''n'') counts the number of totatives of ''n''. The totatives under multiplication modulo ''n'' form the Mult ...

s 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
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and
f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime.
An arithmetic function ''f''(''n'') is ...

, meaning that if two numbers and are relatively prime, then .
This function gives the order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

of the multiplicative group of integers modulo (the group
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ident ...

of unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...

s of the ring
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

$\backslash Z/n\backslash Z$). It is also used for defining the RSA encryption system.
History, terminology, and notation

Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

introduced the function in 1763.Sandifer, p. 203 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
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of m ...

's 1801 treatise ''Disquisitiones Arithmeticae
The (Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to be in relation wit ...

'', 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, 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
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

in common with .
Computing Euler's totient function

There are several formulas for computing .Euler's product formula

It states :$\backslash varphi(n)\; =n\; \backslash prod\_\; \backslash left(1-\backslash frac\backslash right),$ where the product is over the distinctprime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s dividing . (For notation, see Arithmetical function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any Function (mathematics), function ''f''(''n'') whose domain is the natural number, positive integers and whose range is a subset of the complex num ...

.)
An equivalent formulation for $n\; =\; p\_1^\; p\_2^\; \backslash cdots\; p\_r^$, where $p\_1,\; p\_2,\backslash ldots,p\_r$ are the distinct primes dividing ''n'', is:$$\backslash varphi(n)\; =\; p\_1^(p\_11)\backslash ,p\_2^(p\_21)\backslash cdots\; p\_r^(p\_r1).$$The proof of these formulas 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 arecoprime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

to and less than , , , respectively, so that , etc. Then there is a bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

between and by the Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...

.
Value of phi for a prime power argument

If is prime and , then :$\backslash varphi\; \backslash left(p^k\backslash right)\; =\; p^k-p^\; =\; p^(p-1)\; =\; p^k\; \backslash left(\; 1\; -\; \backslash tfrac\; \backslash right).$ ''Proof'': Since is a prime number, the only possible values of are , and the only way to have is if is a multiple of , i.e. , and there are such multiples less than . Therefore, the other numbers are all relatively prime to .Proof of Euler's product formula

Thefundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

states that if there is a unique expression $n\; =\; p\_1^\; p\_2^\; \backslash cdots\; p\_r^,$ where are prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s and each . (The case corresponds to the empty product.) Repeatedly using the multiplicative property of and the formula for gives
:$\backslash begin\; \backslash varphi(n)\&=\&\; \backslash varphi(p\_1^)\backslash ,\; \backslash varphi(p\_2^)\; \backslash cdots\backslash varphi(p\_r^)\backslash \backslash ;\; href="/html/ALL/s/1em.html"\; ;"title="1em">1em$
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 $\backslash $, excluding the sets of integers divisible by the prime divisors.
Example

:$\backslash varphi(20)=\backslash varphi(2^2\; 5)=20\backslash ,(1-\backslash tfrac12)\backslash ,(1-\backslash tfrac15)\; =20\backslash cdot\backslash tfrac12\backslash cdot\backslash tfrac45=8.$ 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:$$\backslash varphi(20)\; =\; \backslash varphi(2^2\; 5^1)=\; 2^(21)\backslash ,5^(51)\; =\; 2\backslash cdot\; 1\backslash cdot\; 1\backslash cdot\; 4\; =\; 8.$$Fourier transform

The totient is thediscrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discret ...

of the , evaluated at 1. Let
:$\backslash mathcal\; \backslash ;\; href="/html/ALL/s/.html"\; ;"title="">$
where for . Then
:$\backslash varphi\; (n)\; =\; \backslash mathcal\; \backslash ;\; href="/html/ALL/s/.html"\; ;"title="">$
The real part of this formula is
:$\backslash varphi\; (n)=\backslash sum\backslash limits\_^n\; \backslash gcd(k,n)\; \backslash cos\; .$
For example, using $\backslash cos\backslash tfrac5\; =\; \backslash tfrac4$ and $\backslash cos\backslash tfrac5\; =\; \backslash tfrac4$:$$\backslash begin\; \backslash varphi(10)\; \&=\&\; \backslash gcd(1,10)\backslash cos\backslash tfrac\; +\; \backslash gcd(2,10)\backslash cos\backslash tfrac\; +\; \backslash gcd(3,10)\backslash cos\backslash tfrac+\backslash cdots+\backslash gcd(10,10)\backslash cos\backslash tfrac\backslash \backslash \; \&=\&\; 1\backslash cdot(\backslash tfrac4)\; +\; 2\backslash cdot(\backslash tfrac4)\; +\; 1\backslash cdot(-\backslash tfrac4)\; +\; 2\backslash cdot(-\backslash tfrac4)\; +\; 5\backslash cdot\; (-1)\; \backslash \backslash \; \&\&\; +\backslash \; 2\backslash cdot(-\backslash tfrac4)\; +\; 1\backslash cdot(-\backslash tfrac4)\; +\; 2\backslash cdot(\backslash tfrac4)\; +\; 1\backslash cdot(\backslash tfrac4)\; +\; 10\; \backslash cdot\; (1)\; \backslash \backslash \; \&=\&\; 4\; .\; \backslash end$$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 :$\backslash sum\_\backslash varphi(d)=n,$ where the sum is over all positive divisors of , can be proven in several ways. (SeeArithmetical function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any Function (mathematics), function ''f''(''n'') whose domain is the natural number, positive integers and whose range is a subset of the complex num ...

for notational conventions.)
One proof is to note that is also equal to the number of possible generators of the cyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

; specifically, if with , then is a generator for every coprime to . Since every element of generates a cyclic subgroup
In group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...

, and all subgroups are 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

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:
:$\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac.$
Put them into lowest terms:
:$\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\; \backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac,\backslash ,\backslash tfrac$
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
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul J ...

applied to the divisor sum formula gives
:$\backslash varphi(n)\; =\; \backslash sum\_\; \backslash mu\backslash left(\; d\; \backslash right)\; \backslash cdot\; \backslash frac\; =\; n\backslash sum\_\; \backslash frac,$
where is the Möbius function
The Möbius function is an important multiplicative function
:''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...

, the multiplicative function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and
f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime.
An arithmetic function ''f''(''n'') is ...

defined by $\backslash mu(p)\; =\; -1$ and $\backslash mu(p^k)\; =\; 0$ for each prime and . This formula may also be derived from the product formula by multiplying out $\backslash prod\_\; (1\; -\; \backslash frac)$ to get $\backslash sum\_\; \backslash frac.$
An example:$$\backslash begin\; \backslash varphi(20)\; \&=\; \backslash mu(1)\backslash cdot\; 20\; +\; \backslash mu(2)\backslash cdot\; 10\; +\backslash mu(4)\backslash cdot\; 5\; +\backslash mu(5)\backslash cdot\; 4\; +\; \backslash mu(10)\backslash cdot\; 2+\backslash mu(20)\backslash cdot\; 1\backslash \backslash ;\; href="/html/ALL/s/5em.html"\; ;"title="5em">5em$$
Some values

The first 100 values are shown in the table and graph below: : In the graph at right the top line is anupper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), sh ...

valid for all other than one, and attained if and only if is a prime number. A simple lower bound is $\backslash varphi(n)\; \backslash ge\; \backslash sqrt$, which is rather loose: in fact, the lower limit of the graph is proportional to .
Euler's theorem

This states that if and arerelatively prime
In number theory, two integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

then
:$a^\; \backslash equiv\; 1\backslash mod\; n.$
The special case where is prime is known as Fermat's little theorem
Fermat's little theorem states that if is a prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not ...

.
This follows from Lagrange's theorem and the fact that is the order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

of the multiplicative group of integers modulo .
The RSA cryptosystem
RSA (Rivest–Shamir–Adleman) is a public-key cryptography, public-key cryptosystem that is widely used for secure data transmission. It is also one of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leona ...

is based on this theorem: it implies that the inverse
Inverse or invert may refer to:
Science and mathematics
* Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence
* Additive inverse (negation), the inverse of a number that, when ad ...

of the function , where is the (public) encryption exponent, is the function , where , the (private) decryption exponent, is the multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

of modulo . The difficulty of computing without knowing the factorization of is thus the difficulty of computing : this is known as the RSA problem
In cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia
''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Gree ...

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

- $a\backslash mid\; b\; \backslash implies\; \backslash varphi(a)\backslash mid\backslash varphi(b)$
- $n\; \backslash mid\; \backslash varphi(a^n-1)\; \backslash quad\; \backslash text\; a,n\; >\; 1$
- $\backslash varphi(mn)\; =\; \backslash varphi(m)\backslash varphi(n)\backslash cdot\backslash frac\; \backslash quad\backslash textd\; =\; \backslash operatorname(m,n)$
Note the special cases

*$\backslash varphi(2m)\; =\; \backslash begin\; 2\backslash varphi(m)\; \&\backslash text\; m\; \backslash text\; \backslash \backslash \; \backslash varphi(m)\; \&\backslash text\; m\; \backslash text\; \backslash end$ *$\backslash varphi\backslash left(n^m\backslash right)\; =\; n^\backslash varphi(n)$ - $\backslash varphi(\backslash operatorname(m,n))\backslash cdot\backslash varphi(\backslash operatorname(m,n))\; =\; \backslash varphi(m)\backslash cdot\backslash varphi(n)$
Compare this to the formula

*$\backslash operatorname(m,n)\backslash cdot\; \backslash operatorname(m,n)\; =\; m\; \backslash cdot\; n$ (Seeleast common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ....) - is even for . Moreover, if has distinct odd prime factors,
- For any and such that there exists an such that .
- $\backslash frac=\backslash frac$
where is the radical of (the product of all distinct primes dividing ).

- $\backslash sum\_\; \backslash frac\; =\; \backslash frac$
- $\backslash sum\_\backslash !\backslash !k\; =\; \backslash tfrac12\; n\backslash varphi(n)\; \backslash quad\; \backslash textn>1$
- $\backslash sum\_^n\backslash varphi(k)\; =\; \backslash tfrac12\; \backslash left(1+\; \backslash sum\_^n\; \backslash mu(k)\backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor^2\backslash right)\; =\backslash frac3n^2+O\backslash left(n(\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac43\backslash right)$ ( cited in)
- $\backslash sum\_^n\backslash frac\; =\; \backslash sum\_^n\backslash frac\backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor=\backslash frac6n+O\backslash left((\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac43\backslash right)$
- $\backslash sum\_^n\backslash frac\; =\; \backslash fracn-\backslash frac2+O\backslash left((\backslash log\; n)^\backslash frac23\backslash right)$
- $\backslash sum\_^n\backslash frac\; =\; \backslash frac\backslash left(\backslash log\; n+\backslash gamma-\backslash sum\_\backslash frac\backslash right)+O\backslash left(\backslash fracn\backslash right)$
(where is the

Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant A mathematical constant is a key whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an ), or by mathema ...). - $\backslash sum\_\backslash stackrel\; \backslash !\backslash !\backslash !\backslash !\; 1\; =\; n\; \backslash frac\; +\; O\; \backslash left\; (\; 2^\; \backslash right\; )$
where is a positive integer and is the number of distinct prime factors of .

Menon's identity

In 1965 P. Kesava Menon proved :$\backslash sum\_\; \backslash !\backslash !\backslash !\backslash !\; \backslash gcd(k-1,n)=\backslash varphi(n)d(n),$ where is the number of divisors of .Formulae involving the golden ratio

Schneider found a pair of identities connecting the totient function, thegolden ratio
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

and the Möbius function
The Möbius function is an important multiplicative function
:''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' ...

. In this section is the totient function, and is the golden ratio.
They are:
:$\backslash phi=-\backslash sum\_^\backslash infty\backslash frac\backslash log\backslash left(1-\backslash frac\backslash right)$
and
:$\backslash frac=-\backslash sum\_^\backslash infty\backslash frac\backslash log\backslash left(1-\backslash frac\backslash right).$
Subtracting them gives
:$\backslash sum\_^\backslash infty\backslash frac\backslash log\backslash left(1-\backslash frac\backslash right)=1.$
Applying the exponential function to both sides of the preceding identity yields an infinite product formula for :
:$e=\; \backslash prod\_^\; \backslash left(1-\backslash frac\backslash right)\backslash !\backslash !^\backslash frac.$
The proof is based on the two formulae
:$\backslash begin\; \backslash sum\_^\backslash infty\backslash frac\backslash left(-\backslash log\backslash left(1-x^k\backslash right)\backslash right)\&=\backslash frac\; \backslash \backslash \; \backslash text\backslash ;\; \backslash sum\_^\backslash infty\backslash frac\backslash left(-\backslash log\backslash left(1-x^k\backslash right)\backslash right)\&=x,\; \backslash qquad\; \backslash quad\; \backslash text\; 01.\; \backslash end\; math>$Generating functions

TheDirichlet series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

for may be written in terms of the Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek alphabet, Greek letter (zeta), is a function (mathematics), mathematical function of a complex variable, and can be expressed as: \zeta(s) = \sum_^\infty \frac = ...

as:
:$\backslash sum\_^\backslash infty\; \backslash frac=\backslash frac.$
The Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a Series (mathematics), series taking the form
:S(q)=\sum_^\infty a_n \frac .
It can be resumed Formal series, formally by expanding the denominator:
:S(q)=\sum_^\infty a_ ...

generating function is
:$\backslash sum\_^\; \backslash frac=\; \backslash frac$
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 :$\backslash lim\backslash sup\; \backslash frac=\; 1,$ but as ''n'' goes to infinity, for all :$\backslash frac\backslash rightarrow\backslash infty.$ 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 :$\backslash frac\; <\; \backslash frac\; <\; 1,$ true for , is proved. We also have :$\backslash lim\backslash inf\backslash frac\backslash log\backslash log\; n\; =\; e^.$ Here isEuler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (letter), gamma ().
It is defined as the limit of a sequence, limiting difference between th ...

, , so and .
Proving this does not quite require the prime number theorem
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

. Since goes to infinity, this formula shows that
:$\backslash lim\backslash inf\backslash frac=\; 0.$
In fact, more is true.
:$\backslash varphi(n)\; >\; \backslash frac\; \backslash quad\backslash text\; n>2$
and
:$\backslash varphi(n)\; <\; \backslash frac\; \backslash quad\backslash text\; n.$
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
In mathematics, the Riemann hypothesis is a conjecture
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

is true, secondly under the contrary assumption."
For the average order, we haveSándor, Mitrinović & Crstici (2006) pp.24–25
:$\backslash varphi(1)+\backslash varphi(2)+\backslash cdots+\backslash varphi(n)\; =\; \backslash frac+O\backslash left(n(\backslash log\; n)^\backslash frac23(\backslash log\backslash log\; n)^\backslash frac43\backslash right)\; \backslash quad\backslash textn\backslash rightarrow\backslash infty,$
due to Arnold Walfisz
Arnold Walfisz (2 July 1892 – 29 May 1962) was a Jewish-Polish mathematician working in analytic number theory.
Life
After the ''Abitur'' in Warsaw (Poland), Arnold Walfisz studied (1909−14 and 1918−21) in Germany at Munich, Berlin, Heidel ...

, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N. M. Korobov
Nikolai Mikhailovich Korobov (russian: Коробов Николай Михайлович; November 23 1917 – October 25 2004) was a Soviet mathematician specializing in number theory and numerical analysis. He is best known for his work in ana ...

(this is currently the best known estimate of this type). The "Big " stands for a quantity that is bounded by a constant times the function of inside the parentheses (which is small compared to ).
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 provedRibenboim, p.38Sándor, Mitrinović & Crstici (2006) p.16 :$\backslash begin\; \backslash lim\backslash inf\; \backslash frac\&=\; 0\; \backslash quad\backslash text\; \backslash \backslash $\lim\sup \frac&= \infty. \end In 1954 Schinzel and strengthened this, proving that the set :$\backslash left\backslash $ isdense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

in the positive real numbers. They also proved that the set
:$\backslash left\backslash $
is dense in the interval (0,1).
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.Guy (2004) p.144 A ''nontotient
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number th ...

'' 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,Sándor & Crstici (2004) p.230 and indeed every positive integer has a multiple which is an even nontotient.
The number of totient numbers up to a given limit is
:$\backslash frace^$
for a constant .
If counted accordingly to multiplicity, the number of totient numbers up to a given limit is
:$\backslash Big\backslash vert\backslash \backslash Big\backslash vert\; =\; \backslash frac\; \backslash cdot\; x\; +\; R(x)$
where the error term is of order at most for any positive .Sándor et al (2006) p.22
It is known that the multiplicity of exceeds infinitely often for any .Sándor et al (2006) p.21Guy (2004) p.145
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 byWacław Sierpiński
Wacław Franciszek Sierpiński (; 14 March 1882 – 21 October 1969) was a Polish people, Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of fu ...

,Sándor & Crstici (2004) p.229 and it had been obtained as a consequence of Schinzel's hypothesis H In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

. 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 .Sándor & Crstici (2004) p.228
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 calledFermat prime
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s, 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
:2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... .
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 (the "encryption key") are released to the public, and (the "decryption key") 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 1932D. H. Lehmer
Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas
__NOTOC__
François ...

asked if there are any composite numbers such that divides . 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 (i.e. ). In 1980 Cohen and Hagis proved that and that . Further, Hagis showed that if 3 divides then and .Guy (2004) p.142
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.See also

*Carmichael function
In number theory, a branch of mathematics, the Carmichael function associates to every positive integer a positive integer , defined as the smallest positive integer such that
:
for every integer between 1 and that is coprime to . In algebraic ...

*Duffin–Schaeffer conjectureThe Duffin–Schaeffer conjecture is a conjecture (now a theorem) in mathematics, specifically, the Diophantine_approximation#Khinchin's_theorem_and_extensions, Diophantine approximation proposed by Richard Duffin, R. J. Duffin and Albert Charles Sc ...

* Generalizations of Fermat's little theorem
*Highly composite number
__FORCETOC__
A highly composite number (sometimes called ''anti-prime'') is a Positive number, positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer ...

* Multiplicative group of integers modulo
*Ramanujan sum
In number theory, Ramanujan's sum, usually denoted ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula
: c_q(n) = \sum_ e^,
where (''a'', ''q'') = 1 means that ''a'' only takes on values coprime ...

*Totient summatory function In number theory, the totient summatory function \Phi(n) is a summatory function of Euler's totient function defined by:
:\Phi(n) := \sum_^n \varphi(k), \quad n\in \mathbf
Properties
Using Möbius inversion formula, Möbius inversion to the tot ...

* Dedekind psi function
Notes

References

The ''Disquisitiones Arithmeticae
The (Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to be in relation wit ...

'' has been translated from Latin into English and German. The German edition includes all of Gauss' papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
References to the ''Disquisitiones'' are of the form Gauss, DA, art. ''nnn''.
*. See paragraph 24.3.2.
*
* Dickson, Leonard Eugene, "History Of The Theory Of Numbers", vol 1, chapter 5 "Euler's Function, Generalizations; Farey Series", Chelsea Publishing 1952
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External links

*Euler's Phi Function and the Chinese Remainder Theorem — proof that is multiplicative

*Dineva, Rosica

The Euler Totient, the Möbius, and the Divisor Functions

*Plytage, Loomis, Polhil

Summing Up The Euler Phi Function

{{Totient Modular arithmetic Multiplicative functions Articles containing proofs Algebra Number theory Leonhard Euler