Math 2.0

Number theory Theorem

Sum of the totient over the divisors

Builds on Euler's totient functionDivisibility

Statement

Let nn be a positive integer. Then ∑d∣nφ(d)=n\sum_{d \mid n} \varphi(d) = n, where the sum runs over the positive divisors dd of nn, and φ\varphi is Euler's totient function.

Remarks

The proof: sort the fractions 1n,2n,…,nn\frac{1}{n}, \frac{2}{n}, \dots, \frac{n}{n} by their denominators in lowest terms. For each divisor dd of nn, exactly φ(d)\varphi(d) of them have the denominator dd. For example, for n=12n = 12: φ(1)+φ(2)+φ(3)+φ(4)+φ(6)+φ(12)=1+1+2+2+2+4=12\varphi(1) + \varphi(2) + \varphi(3) + \varphi(4) + \varphi(6) + \varphi(12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. With the Möbius inversion formula, the result gives φ(n)=∑d∣nμ(d) n/d\varphi(n) = \sum_{d \mid n} \mu(d) \, n / d.

History

Gauss proved this identity in his Disquisitiones Arithmeticae of 1801.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCoprime integers, definition, no evidence levelCoprime integersdefinitionEuler's totient function, definition, no evidence levelEuler's totient funct…definitionSum of the totient over the divisors, statement, no evidence levelSum of the totient ov…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Divisibility definition
  2. Greatest common divisor definition
  3. Coprime integers definition
  4. Euler's totient function definition
  5. Sum of the totient over the divisors statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler%27s_totient_function, Divisor sum
About this page
Tag09JXE6XMCC
URIm2://09JXE6XMCC
Aliasesgauss-totient-sum
Kindstatement, theorem
WritingW0 stub
MSC11A25
StatusThis copy of the site has no build data. Run m2 status 09JXE6XMCC.
UsesEuler's totient function, Divisibility

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