Euler's totient function explained

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

\varphi(n) or \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 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 .

\Z/n\Z ).^[1] It is also used for defining the RSA encryption system.

History, terminology, and notation

Leonhard Euler introduced the function in 1763.^{[2] [3] [4]} 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".^[5] This definition varies from the current definition for the totient function at but is otherwise the same. The now-standard notation^{[3] [6]} comes from Gauss's 1801 treatise Disquisitiones Arithmeticae,^{[7] [8]} 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,^[9] 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

\varphi(n) = n \prod _{p\mid} \left( 1- (1)/(p) \right), where the product is over the distinct prime numbers dividing . (For notation, see Arithmetical function.)

An equivalent formulation is$$\backslash varphi(n)\; =\; p\_1^(p\_11)\backslash ,p\_2^(p\_21)\backslash cdots\; p\_r^(p\_r1),$$ where

n = p ^{k _{1}} _{1} p ^{k _{2}} _{2} … p ^{k _{r}} _{r} is the prime factorization of n (that is, p _{1,} p _{2,\ldots,p} _{r} are distinct prime numbers).

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

\varphi \left(p ^{k\right)} = p ^{k-p} ^{k-1} = p ^{k-1} (p -1 ) = p ^{k} \left( 1 - \tfrac{ 1 }{p} \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, 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

n = p ^{k _{1}} _{1} p ^{k _{2}} _{2} … p ^{k _{r}} _{r} , where are prime numbers and each . (The case corresponds to the empty product.) Repeatedly using the multiplicative property of and the formula for gives\begin{array} {rcl} \varphi(n)& = & \varphi(p ^{k _{1}} _{1} ) \varphi(p ^{k _{2}} _{2} ) … \varphi(p ^{k _{r}} _{r} )\\[. 1 em] & = & p ^{k _{1}} _{1} \left( 1- (1)/(p _{1}) \right) p ^{k _{2}} _{2} \left( 1- (1)/(p _{2}) \right) … p ^{k _{r}} _{r} \left( 1- (1)/(p _{r}) \right)\\[. 1 em] & = & p ^{k _{1}} _{1} p ^{k _{2}} _{2} … p ^{k _{r}} _{r} \left( 1- (1)/(p _{1}) \right) \left( 1- (1)/(p _{2}) \right) … \left( 1- (1)/(p _{r}) \right)\\[. 1 em] & = &n \left( 1- (1)/(p _{1}) \right)\left( 1- (1)/(p _{2}) \right) … \left( 1- (1)/(p _{r}) \right). \end{array}

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

\{ 1 , 2 ,\ldots,n\} , excluding the sets of integers divisible by the prime divisors.

Example

\varphi( 20 ) = \varphi( 2 ^{2} 5 ) =20 ( 1- \tfrac 12 )( 1- \tfrac 15 ) =20 ⋅ \tfrac 12 ⋅ \tfrac 45=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 the discrete Fourier transform of the gcd, evaluated at 1. Let

l{F} \{ x \}[m] = \sum\limits ^{n} _{k =1} x _{k} ⋅ e ^{{-2\pi} (mk)/(n) }

where for . Then

\varphi (n) = l{F} \{ x \}[ 1 ] = \sum\limits ^{n} _{k =1} \gcd(k,n) e ^{-2 \pi i (k)/(n)} .

The real part of this formula is

\varphi (n) = \sum\limits ^{n} _{k =1} \gcd(k,n) \cos {\tfrac{ 2 \pi k}{n}} .

For example, using

\cos\tfrac{\pi} 5 = \tfrac{\sqrt 5+1 } 4 and \cos\tfrac{ 2 \pi} 5 = \tfrac{\sqrt 5-1 } 4 :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,^[10] that

\sum _{d\mid} \varphi(d) = n,

where the sum is over all positive divisors of, can be proven in several ways. (See Arithmetical function for notational conventions.)

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.^[11] 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.^[12] For example, let and consider the positive fractions up to 1 with denominator 20:

\tfrac{ 1 }{ 20 },\tfrac{ 2 }{ 20 },\tfrac{ 3 }{ 20 },\tfrac{ 4 }{ 20 }, \tfrac{ 5 }{ 20 },\tfrac{ 6 }{ 20 },\tfrac{ 7 }{ 20 },\tfrac{ 8 }{ 20 }, \tfrac{ 9 }{ 20 },\tfrac{ 10 }{ 20 },\tfrac{ 11 }{ 20 },\tfrac{ 12 }{ 20 }, \tfrac{ 13 }{ 20 },\tfrac{ 14 }{ 20 },\tfrac{ 15 }{ 20 },\tfrac{ 16 }{ 20 }, \tfrac{ 17 }{ 20 },\tfrac{ 18 }{ 20 },\tfrac{ 19 }{ 20 },\tfrac{ 20 }{ 20 }.

Put them into lowest terms:

\tfrac{ 1 }{ 20 },\tfrac{ 1 }{ 10 },\tfrac{ 3 }{ 20 },\tfrac{ 1 }{ 5 }, \tfrac{ 1 }{ 4 },\tfrac{ 3 }{ 10 },\tfrac{ 7 }{ 20 },\tfrac{ 2 }{ 5 }, \tfrac{ 9 }{ 20 },\tfrac{ 1 }{ 2 },\tfrac{ 11 }{ 20 },\tfrac{ 3 }{ 5 }, \tfrac{ 13 }{ 20 },\tfrac{ 7 }{ 10 },\tfrac{ 3 }{ 4 },\tfrac{ 4 }{ 5 }, \tfrac{ 17 }{ 20 },\tfrac{ 9 }{ 10 },\tfrac{ 19 }{ 20 },\tfrac{ 1 }{ 1 }

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

\varphi(n) = \sum _{d\mid} \mu\left( d \right) ⋅ (n)/(d) = n\sum _{d\mid} (\mu (d))/(d) ,

where is the Möbius function, the multiplicative function defined by

\mu(p) = -1 and \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:

Some values

The first 100 values are shown in the table and graph below:

2	3	4	5	6	7	8	9	10
0	1	1	2	2	4	2	6	4	6	4
10	10	4	12	6	8	8	16	6	18	8
20	12	10	22	8	20	12	18	12	28	8
30	30	16	20	16	24	12	36	18	24	16
40	40	12	42	20	24	22	46	16	42	20
50	32	24	52	18	40	24	36	28	58	16
60	60	30	36	32	48	20	66	32	44	24
70	70	24	72	36	40	36	60	24	78	32
80	54	40	82	24	64	42	56	40	88	24
90	72	44	60	46	72	32	96	42	60	40

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

\varphi(n) \ge \sqrt{n/ 2 } , which is rather loose: in fact, the lower limit of the graph is proportional to .

Euler's theorem

See main article: article and Euler's theorem.

This states that if and are relatively prime then

a ^{\varphi(n)} \equiv 1 \mod n.

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 (public) encryption exponent, is the function, where, the (private) 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 (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\mid b \implies \varphi(a)\mid\varphi(b)
• m \mid \varphi(a ^{m-1)}
• \varphi(mn) = \varphi(m)\varphi(n) ⋅ (d)/(\varphi(d)) where d = \operatorname{gcd}(m,n)

  In particular:

  \varphi( 2 m) = \begin{cases} 2 \varphi(m) & if m iseven \\ \varphi(m) & if m isodd \end{cases} \varphi\left(n ^{m\right)} = n ^{m-1} \varphi(n)
• \varphi(\operatorname{lcm}(m,n)) ⋅ \varphi(\operatorname{gcd}(m,n)) = \varphi(m) ⋅ \varphi(n)

  Compare this to the formula $\backslash operatorname(m,n)\backslash cdot\; \backslash operatorname(m,n)\; =\; m\; \backslash cdot\; n$ (see least common multiple).

• is even for .

  Moreover, if has distinct odd prime factors,

• For any and such that there exists an such that .
• (\varphi(n))/(= n) (\varphi(\operatorname{rad)/((n))}{\operatorname{rad}(n)})

  where is the radical of (the product of all distinct primes dividing).

• \sum _{d} (\mu ^{2(d)})/(\varphi(d)) = (n)/(\varphi(n))  ^[13]
• \sum _{1\le} k = \tfrac 12 n\varphi(n) for n> 1
• \sum ^{n\varphi(k)} _{k =1} = \tfrac 12 \left( 1+ \sum ^{n} _{k =1} \mu(k)\left\lfloor (n)/(k) \right\rfloor ^{2 \right) = (3 {\pi)/(^{2}n})} ^{2+O\left(n(log} n) ^{(23 ( loglog)/(n))} ^{(43 \right))/()}  (^[14] cited in)

• \sum ^{n\varphi(k)} _{k =1} = (3 {\pi)/(^{2}n}) ^{2+O\left(n(log} n) ^{(23 ( loglog)/(n))} ^{(13 \right))/()} [Liu (2016)]

• \sum ^{n (\varphi(k))/(k)} _{k =1} = \sum ^{n (\mu(k))/(k)} _{k =1} \left\lfloor (n)/(\right\rfloor = k) (6 {\pi)/(^{2}n+O\left((log}) n) ^{(23 ( loglog)/(n))} ^{(43 \right))/()}  
• \sum ^{n (k)/(\varphi(k))} _{k =1} = (315 \zeta( 3 ))/(n - 2 \pi ^{4}) (log n)/(2+ O\left(( log) n) ^{(23 \right))/()}  ^[15]
• \sum ^{n (1)/(\varphi(k))} _{k =1} = (315 \zeta( 3 ))/(2 \pi ^{4}) \left( log n + \gamma - \sum _{pprime} (log p)/(\right) + O\left( p ^{2-p+1}) (( log n) ^{(23)/()})/(n\right))  

  (where is the Euler–Mascheroni constant).

Menon's identity

See main article: Menon's identity. In 1965 P. Kesava Menon proved

\sum _{\stackrel{1\le} { \gcd(k,n) =1 }} \gcd(k -1 ,n) = \varphi(n)d(n), where is the number of divisors of .

Divisibility by any fixed positive integer

The following property, which is part of the « folklore » (i.e., apparently unpublished as a specific result: see the introduction of this article in which it is stated as having « long been known ») has important consequences. For instance it rules out uniform distribution of the values of

\varphi(n) in the arithmetic progressions modulo q for any integer q> 1 .

• For every fixed positive integer

q , the relation q|\varphi(n) holds for almost all n , meaning for all but o(x) values of n\le x as x → inf ty .

This is an elementary consequence of the fact that the sum of the reciprocals of the primes congruent to 1 modulo

q 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:

\sum ^{inf ty} _{n =1} (\varphi(n))/(= n ^{s}) (\zeta(s -1 ))/(\zeta(s)) where the left-hand side converges for \Re (s)> 2 .

The Lambert series generating function is

\sum ^{inf ty} _{n =1} (\varphi(n) q ^{n})/(1- q ^{n}) = (q)/(( 1- q) ^{2})

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

\lim\sup (\varphi(n))/(n) = 1 ,

but as n goes to infinity, for all

(\varphi(n))/(n ^{1-\delta}) → inf ty.

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

(6)/(\pi ^{2}) < (\varphi(n) \sigma(n))/(n ^{2}) < 1 ,

true for, is proved.

We also have

\lim inf (\varphi(n))/(n) loglog n = e ^{-\gamma} .

Here is Euler's constant,, so and .

Proving this does not quite require the prime number theorem.^[16] Since goes to infinity, this formula shows that

\lim inf (\varphi(n))/(n) = 0 .

In fact, more is true.^{[17] [18] [19]}

\varphi(n) > (n)/(e ^{\gamma} log log n + (3)/(loglog n)) for n> 2

and

\varphi(n) < (n)/(e loglog n) forinfinitelymany 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 is true, secondly under the contrary assumption."^[19]

For the average order, we have^{[14] [20]}

\varphi( 1 ) + \varphi( 2 ) + … + \varphi(n) = (3 n ^{2})/(\pi ^{2}) + O\left(n( log n) ^{(23 ( loglog)/(n))} ^{(43 \right))/(as n → inf ty,)} 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 (On Euler's function.Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 4, 769–775) improved the error term to O\left(n( log n) ^{(23 ( loglog)/(n))} ^{(13 \right))/()} (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 proved^{[21] [22]}

\begin{align} \lim inf (\varphi(n +1 ))/(\varphi(n)) & = 0 and \\[ 5 px] \lim\sup (\varphi(n +1 ))/(\varphi(n)) & = inf ty. \end{align}

In 1954 Schinzel and Sierpiński strengthened this, proving^{[21] [22]} that the set

\left\{ (\varphi(n +1 ))/(\varphi(n)) , n = 1 , 2 ,\ldots\right\}

is dense in the positive real numbers. They also proved^[21] that the set

\left\{ (\varphi(n))/(n) , n = 1 , 2 ,\ldots\right\}

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.^[23] 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,^[24] and indeed every positive integer has a multiple which is an even nontotient.^[25]

The number of totient numbers up to a given limit is

(x)/(log x) e ^{( C + o( 1 ) ) ( logloglog x) ^{2}}

for a constant .^[26]

If counted accordingly to multiplicity, the number of totient numbers up to a given limit is

\ vert\{ n : \varphi(n) \le x \} \ vert = (\zeta( 2 )\zeta( 3 ))/(\zeta( 6 )) ⋅ x + R(x)

where the error term is of order at most for any positive .^[27]

It is known that the multiplicity of exceeds infinitely often for any .^{[28] [29]}

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,^[30] and it had been obtained as a consequence of Schinzel's hypothesis H.^[26] Indeed, each multiplicity that occurs, does so infinitely often.^{[26] [29]}

However, no number is known with multiplicity . Carmichael's totient function conjecture is the statement that there is no such .^[31]

Perfect totient numbers

See main article: article and Perfect totient number. A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

Applications

Cyclotomy

See main article: article and Constructible polygon.

In the last section of the Disquisitiones^{[32] [33]} Gauss proves^[34] 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^[35]

2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40,... .

Prime number theorem for arithmetic progressions

See main article: article.

The RSA cryptosystem

See main article: article and RSA (algorithm).

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 efficiently factored or if could be efficiently computed without factoring .

Unsolved problems

Lehmer's conjecture

See main article: article and Lehmer's totient problem.

If is prime, then . In 1932 D. H. Lehmer asked if there are any composite numbers such that divides . None are known.^[36]

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 .^[37] Further, Hagis showed that if 3 divides then and .^{[38] [39]}

Carmichael's conjecture

See main article: article and Carmichael's totient function 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.^[23]

Riemann hypothesis

The Riemann hypothesis is true if and only if the inequality

(n)/(\varphi (n)) <e ^{\gamma} loglog n + (e ^{\gamma} ( 4+ \gamma -log4 \pi))/(\sqrt{ log n) }is true for all where is Euler's constant and is the product of the first primes.^[40]

See also

• Carmichael function (λ)
• Dedekind psi function
• Divisor function (σ)
• Duffin–Schaeffer conjecture
• Generalizations of Fermat's little theorem
• Highly composite number
• Multiplicative group of integers modulo
• Ramanujan sum
• Totient summatory function

References

The Disquisitiones Arithmeticae has been translated from Latin into English and German. The German edition includes all of Gauss's 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
• .
  • • • • • .
          • Book: Sándor . Jozsef . Crstici . Borislav . Handbook of number theory II . limited . Dordrecht . Kluwer Academic . 2004 . 1-4020-2546-7 . 1079.11001 . 179–327 .
• .

External links

• • Euler's Phi Function and the Chinese Remainder Theorem — proof that is multiplicative
• Euler's totient function calculator in JavaScript — up to 20 digits
• Dineva, Rosica, The Euler Totient, the Möbius, and the Divisor Functions
• Plytage, Loomis, Polhill Summing Up The Euler Phi Function

Notes and References

1. See Euler's theorem.
2. L. Euler "Theoremata arithmetica nova methodo demonstrata" (An arithmetic theorem proved by a new method), Novi commentarii academiae scientiarum imperialis Petropolitanae (New Memoirs of the Saint-Petersburg Imperial Academy of Sciences), 8 (1763), 74–104. (The work was presented at the Saint-Petersburg Academy on October 15, 1759. A work with the same title was presented at the Berlin Academy on June 8, 1758). Available on-line in: Ferdinand Rudio,, Leonhardi Euleri Commentationes Arithmeticae, volume 1, in: Leonhardi Euleri Opera Omnia, series 1, volume 2 (Leipzig, Germany, B. G. Teubner, 1915), pages 531–555. On page 531, Euler defines as the number of integers that are smaller than and relatively prime to (... aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, ...), which is the phi function, φ(N).
3. Sandifer, p. 203
4. Graham et al. p. 133 note 111
5. L. Euler, Speculationes circa quasdam insignes proprietates numerorum, Acta Academiae Scientarum Imperialis Petropolitinae, vol. 4, (1784), pp. 18–30, or Opera Omnia, Series 1, volume 4, pp. 105–115. (The work was presented at the Saint-Petersburg Academy on October 9, 1775).
6. Both and are seen in the literature. These are two forms of the lower-case Greek letter phi.
7. Gauss, Disquisitiones Arithmeticae article 38
8. Book: Cajori, Florian . Florian Cajori . A History Of Mathematical Notations Volume II . 1929 . Open Court Publishing Company. §409.
9. J. J. Sylvester (1879) "On certain ternary cubic-form equations", American Journal of Mathematics, 2 : 357-393; Sylvester coins the term "totient" on page 361.
10. Gauss, DA, art 39
11. Gauss, DA art. 39, arts. 52-54
12. Graham et al. pp. 134-135
13. Dineva (in external refs), prop. 1
14. Book: Walfisz, Arnold . 0146.06003 . Arnold Walfisz . Weylsche Exponentialsummen in der neueren Zahlentheorie . de . Mathematische Forschungsberichte . 16 . Berlin . . 1963 .
15. R. . Sitaramachandrarao . On an error term of Landau II . Rocky Mountain J. Math. . 15 . 1985 . 2 . 579–588. 10.1216/RMJ-1985-15-2-579 . free .
16. In fact Chebyshev's theorem andMertens' third theorem is all that is needed.
17. Theorem 15 of Rosser . J. Barkley . Schoenfeld . Lowell . Approximate formulas for some functions of prime numbers . Illinois J. Math. . 6 . 1962 . 1 . 64–94 . 10.1215/ijm/1255631807 . free .
18. Bach & Shallit, thm. 8.8.7
19. Book: Ribenboim. The Book of Prime Number Records . 2nd . Springer-Verlag . New York . How are the Prime Numbers Distributed? §I.C The Distribution of Values of Euler's Function . 172–175 . 10.1007/978-1-4684-0507-1_5 . 1989 . 978-1-4684-0509-5 .
20. Sándor, Mitrinović & Crstici (2006) pp.24–25
21. Ribenboim, p.38
22. Sándor, Mitrinović & Crstici (2006) p.16
23. Guy (2004) p.144
24. Sándor & Crstici (2004) p.230
25. 0772.11001 . Zhang . Mingzhi . On nontotients . . 43 . 2 . 168–172 . 1993 . 0022-314X . 10.1006/jnth.1993.1014. free .
26. 0914.11053 . Ford . Kevin . The distribution of totients . Ramanujan J. . Developments in Mathematics . 2 . 1–2 . 67–151 . 1998 . 1382-4090 . 10.1007/978-1-4757-4507-8_8. 1104.3264 . 978-1-4419-5058-1 .
27. Sándor et al (2006) p.22
28. Sándor et al (2006) p.21
29. Guy (2004) p.145
30. Sándor & Crstici (2004) p.229
31. Sándor & Crstici (2004) p.228
32. Gauss, DA. The 7th § is arts. 336–366
33. Gauss proved if satisfies certain conditions then the -gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the -gon is constructible, then must satisfy Gauss's conditions
34. Gauss, DA, art 366
35. Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones
36. Ribenboim, pp. 36–37.
37. 0436.10002 . Cohen . Graeme L. . Hagis . Peter Jr. . On the number of prime factors of if divides . Nieuw Arch. Wiskd. . III Series . 28 . 177–185 . 1980 . 0028-9825 .
38. 0668.10006 . Hagis . Peter Jr. . On the equation . Nieuw Arch. Wiskd. . IV Series . 6 . 3 . 255–261 . 1988 . 0028-9825 .
39. Guy (2004) p.142
40. Book: Broughan . Kevin . Equivalents of the Riemann Hypothesis, Volume One: Arithmetic Equivalents . Cambridge University Press . 2017 . First . 978-1-107-19704-6. Corollary 5.35