Skip to main content

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.

Comments

Post a Comment

Popular posts from this blog

The Tribe of Math Mentors: 11 Questions Every Educator Should Answer

It is easy to get caught up in the vague, existential questions of education: How do I become a better teacher? How do I make math engaging? How do I survive the burnout? But as author Tim Ferriss noted when writing his book Tribe of Mentors, "Life punishes the vague wish and rewards the specific ask." When Ferriss set out to deconstruct the habits of world-class performers, he didn’t ask them broad questions about "the secret to success." He engineered 11 highly specific questions designed to bypass rehearsed answers and force his subjects to share actionable, vulnerable, and unconventional insights. Recently, I started thinking about how perfectly this methodology translates to our world. What if we asked these exact types of questions to master math teachers? What if we used them to guide the next generation of educators? Here is my best thinking of what Ferriss’s Tribe of Mentors questionnaire looks like when translated into the context of the mathematics classr...
Post a Comment
Read more

Beyond Taylor Series: The Magic and History of Padé Approximations

If you have ever taken a calculus class, you probably remember the Taylor series. It is the mathematical magic trick that lets you turn complicated functions—like sines, cosines, and exponentials—into simple, infinitely long polynomials. For centuries, it has been a cornerstone of numerical mathematics. But the Taylor series has a dark secret: it frequently breaks. If a function has a vertical asymptote (a pole) or if you move too far from your starting point, the Taylor series spirals out of control into infinity. It is strictly bounded by what mathematicians call a "radius of convergence." Enter the Padé approximation . Instead of using a single polynomial to estimate a function, a Padé approximant uses a fraction (a ratio of two polynomials). This simple structural change unlocks a profound level of mathematical power, allowing us to see past the limits of Taylor series and model complex, chaotic systems in modern physics and engineering. A Brief History: From Franc...
Post a Comment
Read more

Unlocking the Matrix: A Guide to Solving Systems of 3 Equations

    If you have ever felt like algebra is just moving letters around a page until a number falls out, solving systems of equations in three variables might seem like the ultimate boss battle. Instead of just dealing with x and y on a flat piece of paper, a third variable (like z, or in our case r, s, and t) brings the math into three-dimensional space. Every equation represents a flat plane, and solving the system means finding the exact 3D coordinate where all three planes intersect. But before we dive into a full 3x3 system, let's talk about what happens when the number of equations doesn't perfectly match the number of variables. The Goldilocks Problem: Too Many or Not Enough Equations? For a system to have a single, beautifully neat solution, you generally need exactly as many equations as you have variables. But what if the balance is off? Not Enough Equations (Underdetermined Systems): Let's say you have 3 variables but only 2 equations. This is like having two plane...
Post a Comment
Read more