Euler's totient function

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For other functions named after Euler, see List of things named after Leonhard Euler. For other functions named phi, see phi.
The first thousand values of \varphi(n)

In number theory, Euler's totient or phi function, φ(n), is an arithmetic function that counts the totatives of n, that is, the positive integers less than or equal to n that are relatively prime to n. Thus, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ kn for which gcd(n, k) = 1.[1][2] The totient function is a multiplicative function, meaning that if two numbers m and n are relatively prime (to each other), then φ(mn) = φ(m)φ(n).[3][4]

For example let n = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤ k ≤ 9, that is 1, 2, 4, 5, 7 and 8 are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.

The totient function is important mainly because it gives the order of the multiplicative group of integers modulo n (the group of units of the ring \mathbb{Z}/n\mathbb{Z}). See Euler's theorem. The totient function also plays a key role in the definition of the RSA encryption system.

History, terminology, and notation[edit]

Leonhard Euler introduced the function in 1763.[5][6][7] 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 π to denote it: he wrote πD for "the multitude of numbers less than D, and which have no common divisor with it".[8] The standard notation[6][9] φ(A) comes from Gauss' 1801 treatise Disquisitiones Arithmeticae.[10] However, Gauss doesn't use parentheses around the argument and writes φA. Thus, it is usually called Euler's phi function or simply the phi function.

In 1879 J. J. Sylvester coined the term totient for this function,[11][12] so it is also referred to as the totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's.

The cototient of n is defined as n – φ(n), i.e. the number of positive integers less than or equal to n that are divisible by at least one prime that also divides n.

Computing Euler's function[edit]

There are several formulae for the totient.

Euler's product formula[edit]

It states


\varphi(n) =n \prod_{p\mid n} \left(1-\frac{1}{p}\right),

where the product is over the distinct prime numbers dividing n. (The notation is described in the article Arithmetical function.)

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

φ(n) is multiplicative[edit]

This means that if gcd(m, n) = 1, then φ(mn) = φ(m) φ(n). (Sketch of proof: let A, B, C be the sets of residue classes modulo-and-coprime to m, n, mn respectively; then there is a bijection between A × B and C, by the Chinese remainder theorem.)

φ(pk) = pkpk−1 = pk−1(p − 1)[edit]

That is, if p is prime and k ≥ 1 then

\varphi(p^k) = p^k -p^{k-1} =p^{k-1}(p-1) = p^k \left( 1 - \frac{1}{p} \right).

Proof: Since p is a prime number the only possible values of gcd(pk, m) are 1, p, p2, ..., pk, and the only way for gcd(pk, m) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to pk are p, 2p, 3p, ..., pk − 1p = pk, and there are pk − 1 of them. Therefore the other pkpk − 1 numbers are all relatively prime to pk.

Proof of the formula: The fundamental theorem of arithmetic states that if n > 1 there is a unique expression for n,

n = p_1^{k_1} \cdots p_r^{k_r},

where p1 < p2 < ... < pr are prime numbers and each ki ≥ 1. (The case n = 1 corresponds to the empty product.)

Repeatedly using the multiplicative property of φ and the formula for φ(pk) gives


\begin{align} 
\varphi(n)
&= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})\\

&=  p_1^{k_1} \left(1- \frac{1}{p_1} \right) p_2^{k_2} \left(1- \frac{1}{p_2} \right) \cdots p_r^{k_r} \left(1- \frac{1}{p_r} \right)\\

&= p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r} \left(1- \frac{1}{p_1} \right) \left(1- \frac{1}{p_2} \right) \cdots \left(1- \frac{1}{p_r} \right)\\

&=n \left(1- \frac{1}{p_1} \right)\left(1- \frac{1}{p_2} \right) \cdots\left(1- \frac{1}{p_r} \right).
\end{align}

This is Euler's product formula.

Example[edit]

\varphi(36)=\varphi\left(2^2 3^2\right)=36\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)=36\cdot\frac{1}{2}\cdot\frac{2}{3}=12.

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[edit]

The totient is the discrete Fourier transform of the gcd, evaluated at 1:   (Schramm (2008))

\begin{align}
  \mathcal{F} \left\{ \mathbf{x} \right\}[m] &= \sum\limits_{k=1}^n x_k \cdot e^{{-2\pi i}\frac{mk}{n}}, \mathbf{x_k} = \left\{ \gcd(k,n) \right \} \quad\text{for}\, k \in \left\{1 \dots n \right\} \\
                                 \varphi (n) &= \mathcal{F} \left\{ \mathbf{x} \right\}[1] = \sum\limits_{k=1}^n \gcd(k,n) e^{{-2\pi i}\frac{k}{n}}.
\end{align}

The real part of this formula is

\varphi (n)=\sum\limits_{k=1}^n \gcd(k,n) \cos {2\pi\frac{k}{n}}.

Note that unlike the other two formulae (the Euler product and the divisor sum) this one does not require knowing the factors of n. However, it does involve the calculation of the greatest common divisor of n and every positive integer less than n, which would suffice to provide the factorization anyway.

Divisor sum[edit]

The property established by Gauss,[13] that


\sum_{d\mid n}\varphi(d)=n,

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

One way is to note that φ(d) is also equal to the number of possible generators of the cyclic group Cd; specifically, if Cd = <g>, then gk is a generator for every k coprime to d. Since every element of Cn generates a cyclic subgroup, and all φ(d) subgroups of CdCn are generated by some element of Cn, the formula follows.[14] 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 nth roots of unity.

This formula can also be derived in a more concrete manner.[15] Let n = 20 and consider the fractions between 0 and 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}

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  \left(\tfrac{ 1}{20},\,\tfrac{ 3}{20},\,\tfrac{ 7}{20},\,\tfrac{ 9}{20},\,\tfrac{ 11}{20},\,\tfrac{13}{20},\,\tfrac{17}{20},\,\tfrac{19}{20}\right). By definition this is φ(20) fractions. Similarly, there are φ(10) = 4 fractions with denominator 10  \left(\tfrac{ 1}{10},\,\tfrac{ 3}{10},\,\tfrac{ 7}{10},\,\tfrac{ 9}{10}\right), φ(5) = 4 fractions with denominator 5  \left(\tfrac{ 1}{5},\,\tfrac{ 2}{5},\,\tfrac{ 3}{5},\,\tfrac{ 4}{5}\right), and so on. Since the same argument works for any number, not just 20, the formula is established.

Möbius inversion gives


  \varphi(n) = \sum_{d\mid n} d \cdot \mu\left(\frac{n}{d} \right) 
             = n\sum_{d\mid n} \frac{\mu (d)}{d},

where μ is the Möbius function.

This formula may also be derived from the product formula by multiplying out    \prod_{p\mid n} \left(1 - \frac{1}{p}\right)   to get    \sum_{d\mid n} \frac{\mu (d)}{d}.

Riemann zeta function limit[edit]

For  n>1 the Euler totient function can be calculated as a limit involving the Riemann zeta function:[citation needed]

 \varphi(n)=n\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \mu(d)(e^{1/d})^{(s-1)}

where

 \zeta(s) is the Riemann zeta function,  \mu is the Möbius function,  e is e (mathematical constant), and  d is a divisor.

Some values of the function[edit]

The first 99 values (sequence A000010 in OEIS) are shown in the table and graph below:[16]

Graph of the first 100 values
\varphi(n) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9
0+   1 1 2 2 4 2 6 4 6
10+ 4 10 4 12 6 8 8 16 6 18
20+ 8 12 10 22 8 20 12 18 12 28
30+ 8 30 16 20 16 24 12 36 18 24
40+ 16 40 12 42 20 24 22 46 16 42
50+ 20 32 24 52 18 40 24 36 28 58
60+ 16 60 30 36 32 48 20 66 32 44
70+ 24 70 24 72 36 40 36 60 24 78
80+ 32 54 40 82 24 64 42 56 40 88
90+ 24 72 44 60 46 72 32 96 42 60

The top line in the graph, y = n − 1, is a true upper bound. It is attained whenever n 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 n/log log n rather than being linear.[17]

Euler's theorem[edit]

Main article: Euler's theorem

This states that if a and n are relatively prime then

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

The special case where n is prime is known as Fermat's little theorem.

This follows from Lagrange's theorem and the fact that φ(n) is the order of the multiplicative group of integers modulo n.

The RSA cryptosystem is based on this theorem: it implies that the inverse of the function a\mapsto a^d \mod n is the function b\mapsto b^e \mod n, where  e is the multiplicative inverse of  d modulo \varphi(n). The difficulty of decoding without knowing the private key is thus the difficulty of computing  e, which is the same as factoring n.

Other formulae involving φ[edit]

  • a\mid b implies \varphi(a)\mid\varphi(b).
  •  n \mid \varphi(a^n-1)       (a, n > 1)
  • \varphi(mn) = \varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)}       where d = gcd(m, n). Note the special cases
  • 
\varphi(2m) = 
\begin{cases}
2\varphi(m) &\text{ if } m \text{ is even} \\
\varphi(m)   &\text{ if } m \text{ is odd}
\end{cases}

and

  • \;\varphi\left(n^m\right) = n^{m-1}\varphi(n).
  • \varphi(\mathrm{lcm}(m,n))\cdot\varphi(\mathrm{gcd}(m,n)) = \varphi(m)\cdot\varphi(n).      
Compare this to the formula       \mathrm{lcm}(m,n)\cdot \mathrm{gcd}(m,n) = m \cdot n.       (See lcm).
  • \varphi(n)\; is even for n \geq 3. Moreover, if n has r distinct odd prime factors, 2^r \mid \varphi(n).
  • For any a > 1 and n > 6 such that  4 \nmid n there exists an  l \geq 2n such that  l \mid \varphi(a^n-1).
  • \sum_{1\le k\le n \atop (k,n)=1}\!\!k = \frac{1}{2}n\varphi(n)\text{ for }n>1
  • \sum_{k=1}^n\varphi(k) = \frac{1}{2}\left(1+ \sum_{k=1}^n \mu(k)\left\lfloor\frac{n}{k}\right\rfloor^2\right)

(here γ is the Euler constant).

  • \sum_{1\le k\le n \atop (k,m)=1} 1 = n \frac {\varphi(m)}{m} + 
O \left ( 2^{\omega(m)} \right ),

where m > 1 is a positive integer and ω(m) is the number of distinct prime factors of m. (a, b) is a standard abbreviation for gcd(a, b).[22]

Menon's identity[edit]

Main article: Menon's identity

In 1965 P. Kesava Menon proved


\sum_{\stackrel{1\le k\le n}{ \gcd(k,n)=1}} \gcd(k-1,n)
=\varphi(n)d(n),

where d(n) = σ0(n) is the number of divisors of n.

Formulae involving the golden ratio[edit]

Schneider[23] found a pair of identities connecting the totient function, the golden ratio and the Möbius function \mu(n). In this section \varphi(n) is the totient function, and 
\phi = \frac{1+\sqrt{5}}{2}= 1.618\dots
is the golden ratio.

They are:


\phi=-\sum_{k=1}^\infty\frac{\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)

and


\frac{1}{\phi}=-\sum_{k=1}^\infty\frac{\mu(k)}{k}\log\left(1-\frac{1}{\phi^k}\right).

Subtracting them gives


\sum_{k=1}^\infty\frac{\mu(k)-\varphi(k)}{k}\log\left(1-\frac{1}{\phi^k}\right)=1.

Applying the exponential function to both sides of the preceding identity yields an infinite product formula for Euler's constant e

e= \prod_{k=1}^{\infty} \left(1-\frac{1}{\phi^k}\right)^\frac{\mu(k)-\varphi(k)}{k}.

The proof is based on the formulae


\sum_{k=1}^\infty\frac{\varphi(k)}{k}(-\log(1-x^k))=\frac{x}{1-x}
    and     
\sum_{k=1}^\infty\frac{\mu(k)}{k}(-\log(1-x^k))=x, 
    valid for 0 < x < 1.

Generating functions[edit]

The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as:[24]

\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)}.

The Lambert series generating function is[25]

\sum_{n=1}^{\infty} \frac{\varphi(n) q^n}{1-q^n}= \frac{q}{(1-q)^2}

which converges for |q| < 1.

Both of these are proved by elementary series manipulations and the formulae for φ(n).

Growth of the function[edit]

In the words of Hardy & Wright, the order of φ(n) is "always ‘nearly n’."[26]

First[27]


\lim\sup \frac{\varphi(n)}{n}= 1,

but as n goes to infinity,[28] for all δ > 0


\frac{\varphi(n)}{n^{1-\delta}}\rightarrow\infty.

These two formulae can be proved by using little more than the formulae for φ(n) and the divisor sum function σ(n).

In fact, during the proof of the second formula, the inequality


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

true for n > 1, is proven.

We also have[17]


\lim\inf\frac{\varphi(n)}{n}\log\log n = e^{-\gamma}.

Here γ is Euler's constant,   γ = 0.577215665...,   eγ = 1.7810724...,   e−γ = 0.56145948... .

Proving this doesn't quite require the prime number theorem.[29][30] Since log log (n) goes to infinity, this formula shows that


\lim\inf\frac{\varphi(n)}{n}= 0.

In fact, more is true.[31][32]


\varphi(n) > \frac {n} {e^\gamma\; \log \log n + \frac {3} {\log \log n}} 
      for n > 2, and

\varphi(n) < \frac  {n} {e^{ \gamma}\log \log n} 
                      for infinitely many n.

Concerning the second inequality, 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."[32]

For the average order, we have[19][33]


\varphi(1)+\varphi(2)+\cdots+\varphi(n) = \frac{3n^2}{\pi^2}+O\left(n(\log n)^{2/3}(\log\log n)^{4/3}\right)\ \ (n\rightarrow\infty),

due to Arnold Walfisz, its proof exploiting estimates on exponential sums due to I. M. Vinogradov and N.M. Korobov (this is currently the best known estimate of this type). The "Big O" stands for a quantity that is bounded by a constant times the function of "n" inside the parentheses (which is small compared to n2).

This result can be used to prove[34] that the probability of two randomly chosen numbers being relatively prime is  \tfrac{6}{\pi^2}.

Ratio of consecutive values[edit]

In 1950 Somayajulu proved[35][36]


\lim\inf \frac{\varphi(n+1)}{\varphi(n)}= 0
        and        

\lim\sup \frac{\varphi(n+1)}{\varphi(n)}= \infty.

In 1954 Schinzel and Sierpiński strengthened this, proving[35][36] that the set


\left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n = 1,2,\cdots\right\}

is dense in the positive real numbers. They also proved[35] that the set


\left\{\frac{\varphi(n)}{n},\;\;n = 1,2,\cdots\right\}

is dense in the interval (0, 1).

Totient numbers[edit]

A totient number is a value of Euler's totient function: that is, an m for which there is at least one x for which φ(x) = m. The valency or multiplicity of a totient number m is the number of solutions to this equation.[37] A nontotient is a natural number which is not a totient number: there are infinitely many nontotients,[38] and indeed every odd number has a multiple which is a nontotient.[39]

The number of totient numbers up to a given limit x is

\frac{x}{\log x}\exp\left({ (C+o(1))(\log\log\log x)^2 }\right) \

for a constant C = 0.8178146... .[40]

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

\vert\{ n : \phi(n) \le x \}\vert = \frac{\zeta(2)\zeta(3)}{\zeta(6)} \cdot x + R(x) \

where the error term R is of order at most x / (\log x)^k for any positive k.[41]

It is known that the multiplicity of m exceeds mδ infinitely often for any δ < 0.55655.[42][43]

Ford's theorem[edit]

Ford (1999) proved that for every integer k ≥ 2 there is a totient number m of multiplicity k: that is, for which the equation φ(x) = m has exactly k solutions; this result had previously been conjectured by Wacław Sierpiński,[44] and it had been obtained as a consequence of Schinzel's hypothesis H.[40] Indeed, each multiplicity that occurs, does so infinitely often.[40][43]

However, no number m is known with multiplicity k = 1. Carmichael's totient function conjecture is the statement that there is no such m.[45]

Applications[edit]

Cyclotomy[edit]

Main article: Constructible polygon

In the last section of the Disquisitiones[46][47] Gauss proves[48] that a regular n-gon can be constructed with straightedge and compass if φ(n) is a power of 2. If n is a power of an odd prime number the formula for the totient says its totient can be a power of two only if a) n is a first power and b) n − 1 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 n-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 n are[49] 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, ... .     (sequence A003401 in OEIS)

The RSA cryptosystem[edit]

Main article: RSA (algorithm)

Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private.

A message, represented by an integer m, where 0 < m < n, is encrypted by computing S = me (mod n).

It is decrypted by computing t = Sd (mod n). Euler's Theorem can be used to show that if 0 < t < n, then t = m.

The security of an RSA system would be compromised if the number n could be factored or if φ(n) could be computed without factoring n.

Unsolved problems[edit]

Lehmer's conjecture[edit]

If p is prime, then φ(p) = p − 1. In 1932 D. H. Lehmer asked if there are any composite numbers n such that φ(n) | n − 1. None are known.[50]

In 1933 he proved that if any such n exists, it must be odd, square-free, and divisible by at least seven primes (i.e. ω(n) ≥ 7). In 1980 Cohen and Hagis proved that n > 1020 and that ω(n) ≥ 14.[51] Further, Hagis showed that if 3 divides n then n > 101937042 and ω(n) ≥ 298848.[52][53]

Carmichael's conjecture[edit]

This states that there is no number n with the property that for all other numbers m, mn, φ(m) ≠ φ(n). 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.[37]

See also[edit]

Notes[edit]

  1. ^ Long (1972, p. 85)
  2. ^ Pettofrezzo & Byrkit (1970, p. 72)
  3. ^ Long (1972, p. 162)
  4. ^ Pettofrezzo & Byrkit (1970, p. 80)
  5. ^ 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, ed., 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 n as the number of integers that are smaller than N and relatively prime to N (… aequalis sit multitudini numerorum ipso N minorum, qui simul ad eum sint primi, …), which is the totient function, φ(N).
  6. ^ a b Sandifer, p. 203
  7. ^ Graham et al. p. 133 note 111
  8. ^ 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).
  9. ^ Both \varphi(n)\; and \phi(n)\; are seen in the literature. These are two forms of the lower-case Greek letter phi.
  10. ^ Gauss, Disquisitiones Arithmeticae article 38
  11. ^ 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.
  12. ^ "totient". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989. 
  13. ^ Gauss, DA, art 39
  14. ^ Gauss, DA art. 39, arts. 52-54
  15. ^ Graham et al. pp. 134-135
  16. ^ The cell for n = 0 in the upper-left corner of the table is empty, as the function φ(n) is commonly defined only for positive integers, so it is not defined for n = 0.
  17. ^ a b Hardy & Wright 1979, thm. 328
  18. ^ Dineva (in external refs), prop. 1
  19. ^ a b Walfisz, Arnold (1963). Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte (in German) 16. Berlin: VEB Deutscher Verlag der Wissenschaften. Zbl 0146.06003. 
  20. ^ R. Sitaramachandrarao. On an error term of Landau II, Rocky Mountain J. Math. 15 (1985), 579-588
  21. ^ Also R. Sitaramachandrarao (loc. cit.)
  22. ^ Bordellès in the external links
  23. ^ All formulae in the section are from Schneider (in the external links)
  24. ^ Hardy & Wright 1979, thm. 288
  25. ^ Hardy & Wright 1979, thm. 309
  26. ^ Hardy & Wright 1979, intro to § 18.4
  27. ^ Hardy & Wright 1979, thm. 326
  28. ^ Hardy & Wright 1979, thm. 327
  29. ^ In fact Chebychev's theorem (Hardy & Wright 1979, thm.7) and Mertens' third theorem is all that's needed
  30. ^ Hardy & Wright 1979, thm. 436
  31. ^ Bach & Shallit, thm. 8.8.7
  32. ^ a b Ribenboim, The Book of Prime Number Records, Section 4.I.C
  33. ^ Sándor, Mitrinović & Crstici (2006) pp.24-25
  34. ^ Hardy & Wright 1979, thm. 332
  35. ^ a b c Ribenboim, p.38
  36. ^ a b Sándor, Mitrinović & Crstici (2006) p.16
  37. ^ a b Guy (2004) p.144
  38. ^ Sándor & Crstici (2004) p.230
  39. ^ Zhang, Mingzhi (1993). "On nontotients". Journal of Number Theory 43 (2): 168–172. doi:10.1006/jnth.1993.1014. ISSN 0022-314X. Zbl 0772.11001. 
  40. ^ a b c Ford, Kevin (1998). "The distribution of totients". Ramanujan J. 2 (1-2): 67–151. doi:10.1007/978-1-4757-4507-8_8. ISSN 1382-4090. Zbl 0914.11053. 
  41. ^ Sándor et al (2006) p.22
  42. ^ Sándor et al (2006) p.21
  43. ^ a b Guy (2004) p.145
  44. ^ Sándor & Crstici (2004) p.229
  45. ^ Sándor & Crstici (2004) p.228
  46. ^ Gauss, DA. The 7th § is arts. 336-366
  47. ^ Gauss proved if n satisfies certain conditions then the n-gon can be constructed. In 1837 Pierre Wantzel proved the converse, if the n-gon is constructible, then n must satisfy Gauss's conditions
  48. ^ Gauss, DA, art 366
  49. ^ Gauss, DA, art. 366. This list is the last sentence in the Disquisitiones
  50. ^ Ribenboim, pp. 36-37.
  51. ^ Cohen, Graeme L.; Hagis, Peter, jun. (1980). "On the number of prime factors of n if φ(n) divides n−1". Nieuw Arch. Wiskd., III. Ser. 28: 177–185. ISSN 0028-9825. Zbl 0436.10002. 
  52. ^ Hagis, Peter, jun. (1988). "On the equation M⋅φ(n)=n−1". Nieuw Arch. Wiskd., IV. Ser. 6 (3): 255–261. ISSN 0028-9825. Zbl 0668.10006. 
  53. ^ Guy (2004) p.142

References[edit]

The Disquisitiones Arithmeticae 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.

External links[edit]