Section 1.1: Understanding Arithmetic Functions

To begin, we must familiarize ourselves with arithmetic functions, which are functions mapping natural numbers to complex numbers. These are defined as f(n) for each natural number n, although the results often yield natural numbers. An arithmetic function can also be represented as a sequence of complex numbers.

Some noteworthy examples include the sum of divisors function σ, Euler's totient function ϕ, harmonic numbers, and the prime counting function π (not to be confused with the mathematical constant). From this point onward, the symbols n, m, and k will denote natural numbers unless stated otherwise.

Section 1.2: Exploring Multiplicative Functions

The concept of arithmetic functions encompasses a wide array of possibilities. Notably, certain properties of these functions are crucial. For instance, the greatest common divisor (gcd) of two numbers n and m is defined as the largest natural number that divides both, such that gcd(12, 18) = 6.

An arithmetic function f is termed multiplicative if for any pair of coprime numbers n and m (where gcd(n, m) = 1), the relationship f(nm) = f(n)f(m) holds. If this is true for all n and m, then f is referred to as completely multiplicative. Such functions are fundamental in various aspects of number theory.

The family of "sum of divisors functions" is particularly significant, which we will revisit shortly.

Section 1.3: Summation Over Divisors

This theory revolves around summing the values of arithmetic functions over the divisors of a number n. We denote d|n to mean "d divides n". Given an arithmetic function g, we can define a new function f based on g, where the summation is taken over the divisors of n. For example, f(8) = g(1) + g(2) + g(4) + g(8) because the divisors of 8 are 1, 2, 4, and 8.

A remarkable property is that if g is multiplicative, then f is also multiplicative. To understand this, consider n and m as coprime natural numbers (gcd(n, m) = 1). The proof involves using the definitions of f and g along with properties of divisors.

This connection between f and g raises many intriguing questions, such as identifying known functions that exist in such pairs or exploring how to express g in terms of f.

Section 2.1: Defining Key Functions

We introduce important yet straightforward arithmetic functions:

• The identity function I is defined as:
  • I(1) = 1, for n > 1, I(n) = 0.

This function plays a vital role in the next sections, explaining its identity nature.

• The unit function u is defined as:
  • u(n) = 1 for all n.
• The function N is defined as:
  • N(n) = n for all n.

Section 2.2: The Euler Totient Function ϕ

The Euler totient function, denoted by ϕ, has been a focal point of study for centuries. It counts the natural numbers less than n that are coprime to n. For example:

• ϕ(1) = 1
• ϕ(6) = 2
• ϕ(7) = 6
• ϕ(10) = 4

In general, for a prime p, ϕ(p) = p - 1. This function possesses several crucial properties in number theory and group theory, and it is also multiplicative.

Section 2.3: The Möbius Function μ

The Möbius function is pivotal in analytic number theory and plays a central role in the study of arithmetic functions. Defined as follows:

• μ(1) = 1. For a natural number n, if n is divisible by a square m² for some m, then μ(n) = 0. If no square divides n (n is square-free), then:
  • μ(n) = -1 if the number of prime factors is odd,
  • μ(n) = 1 if the number of prime factors is even.

The Möbius function is also multiplicative, with initial values including:

• μ(1) = 1, μ(2) = -1, μ(3) = -1, μ(4) = 0, μ(5) = -1, μ(6) = 1.

Section 3.1: Introduction to Dirichlet Convolutions

With these foundations laid, we can now delve into the world of arithmetic functions equipped with the natural operation known as Dirichlet convolution. The convolution of two arithmetic functions f and g is defined as f * g. Notably, this operation is symmetric, meaning f * g = g * f.

We also have:

• (f * g) * h = f * (g * h),
• f * I = I * f = f,

where I is the identity function. This symmetry raises the question of which arithmetic functions have inverses concerning Dirichlet convolution.

It turns out that if f(1) ≠ 0, then f possesses an inverse, leading to a group structure among invertible arithmetic functions. This relationship is fundamental to the theory, although we will not explore it in depth here.

The Möbius function μ and the unit function u are inverses, as μ * u = I.

Section 3.2: The Möbius Inversion Formula

A crucial theorem in number theory states:

Let f and g be arithmetic functions. If f = g * u for all n, then g = f * μ for all n.

This elegant proof utilizes our powerful notation, demonstrating the beauty and simplicity of the result.

Section 3.3: Dirichlet Series and Their Importance

One aspect not yet covered is the origin of Dirichlet convolutions and their significance, particularly in the context of Dirichlet series, which take the form:

[ D(s) = sum_{n=1}^{infty} frac{f(n)}{n^s} ]

where f is an arithmetic function and s is a complex variable.

Dirichlet convolutions appear naturally in the study of these series. Notably, the series where the arithmetic function is the unit function corresponds to the well-known Riemann zeta function ζ(s) for Re(s) > 1.

The first video, "The Queen of Mathematics - Professor Raymond Flood," explores the beauty and impact of number theory through its historical context and applications.

The second video, "The Queen of Mathematics," delves into the significance of number theory and its foundational role in modern mathematics.