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The Hidden Connections Between the Totient, Sigma, Tau, Möbius, and Dirichlet Convolution

The Hidden Connections Between Totient, Sigma, Tau, Möbius, and Dirichlet Convolution

In number theory, certain functions show how integers interact with their divisors. This post introduces five of them — the Euler totient function phi(n), the sum of divisors sigma(n), the number of divisors tau(n), the Möbius function mu(n), and the Dirichlet convolution — using one example number:

Example number: n = 12

1) Euler’s Totient Function φ(n)

Idea: phi(n) counts how many numbers from 1 to n are coprime to n (that is, share no common factors with n except 1).

Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then

phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk)

Example with n = 12: 12 = 2^2 * 3^1

phi(12) = 12 * (1 - 1/2) * (1 - 1/3) = 12 * (1/2) * (2/3) = 4

The numbers 1, 5, 7, and 11 are coprime to 12, so phi(12) = 4.

2) The Sigma Function σ(n)

Idea: sigma(n) is the sum of all positive divisors of n.

Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then

sigma(n) = [(p1^(a1+1) - 1) / (p1 - 1)] * [(p2^(a2+1) - 1) / (p2 - 1)] * ... * [(pk^(ak+1) - 1) / (pk - 1)]

Example with n = 12:

sigma(12) = [(2^3 - 1) / (2 - 1)] * [(3^2 - 1) / (3 - 1)] = (7) * (4) = 28

The divisors of 12 are 1, 2, 3, 4, 6, 12, and their sum is 28. So sigma(12) = 28.

3) The Tau Function τ(n)

Idea: tau(n) counts how many positive divisors n has.

Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then

tau(n) = (a1 + 1) * (a2 + 1) * ... * (ak + 1)

Example with n = 12: 12 = 2^2 * 3^1 tau(12) = (2 + 1) * (1 + 1) = 3 * 2 = 6

The divisors of 12 are 1, 2, 3, 4, 6, and 12. There are 6 in total, so tau(12) = 6.

4) The Möbius Function μ(n)

Idea: mu(n) helps detect whether a number has repeated prime factors.

Definition:

  • mu(1) = 1
  • mu(n) = 0 if n has a squared prime factor
  • mu(n) = (-1)^k if n is a product of k distinct primes

Example with n = 12: 12 = 2^2 * 3. Because 12 has a squared factor (2^2), mu(12) = 0.

5) Möbius Inversion Formula

Idea: Sometimes one function (g) is the sum of another function (f) over the divisors of n. If g(n) = Σ f(d) for all d dividing n, the Möbius inversion formula lets us recover f from g.

Formula:

f(n) = Σ mu(d) * g(n / d), summed over all d dividing n.

This means mu acts like an “inverse” for divisor sums.

6) Dirichlet Convolution

Definition: For two arithmetic functions f and g, their Dirichlet convolution is defined as:

(f * g)(n) = Σ f(d) * g(n / d), summed over all d dividing n.

Examples of how these functions relate:

sigma(n) = id * 1 tau(n) = 1 * 1 phi(n) = id * mu

where "id(n)" means the identity function id(n) = n, and "1(n)" means the constant function that always equals 1.

7) Checking the Relationships for n = 12

Check that sigma = id * 1. Divisors of 12: 1, 2, 3, 4, 6, 12.

(id * 1)(12) = 1*1 + 2*1 + 3*1 + 4*1 + 6*1 + 12*1 = 28 That matches sigma(12) = 28.

Check that phi = id * mu. We will make a small table:

d12/dmu(12/d)d * mu(12/d)
11200
2612
3400
43-1-4
62-1-6
121112

Sum = 0 + 2 + 0 - 4 - 6 + 12 = 4 That matches phi(12) = 4.

8) Summary Table

FunctionMeaningFormulan=12
phi(n)Count of numbers coprime to nphi(n) = n * Π (1 - 1/p)4
sigma(n)Sum of divisorssigma(n) = Π [(p^(a+1) - 1)/(p - 1)]28
tau(n)Number of divisorstau(n) = Π (a + 1)6
mu(n)Detects squared primesmu(1)=1; mu(n)=0 if any p^2 divides n; else (-1)^k0

These relationships show how number theory ties together through divisor functions. The Möbius function “undoes” divisor sums, and the Dirichlet convolution “combines” functions in a structured way. Even with n = 12, we can see the elegance in how these mathematical ideas connect.

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