Chapter 3: Arithmetic Functions

Chapter 3 develops the concept of an arithmetic function (pronounced "air - ith - meh - tick"), which we motivated by studying Euler's $\phi$ function. These functions are some of the most important — and most studied — in all of number theory.

Lecture 11: Euler's Phi Function, Cont'd.
In today's class we began by talking about a few "magic tricks" one can do using modular arithmetic. These were motivated by a posting from the forum which showed a video of a "mathemagician" at work. Afterwards we returned to our discussion of Euler's $\phi$ function from Friday. We first calculated the value of the $\phi$ function for "special integers," and eventually noticed that the $\phi$ function obeys some rules which make the calculation of $\phi(n)$ easy — at least when you have a prime factorization of n.
Lecture 12: Arithmetic Functions
Today we talked a little more about the Euler $\phi$ function, then moved on to talk about arithmetic functions more generally. We saw a few examples of arithmetic functions, but we spent the majority of our time discussing the properties these functions have — especially when they respect multiplication.
Lecture 13: Counting Divisors
In today's class we started by giving an alternate proof of the fact that $\sum_{d \mid n} \phi(d) = n$. We then discussed the function $\nu(n)$, which counts the number of positive divisors of n. We saw that the function is multiplicative (but not completely multiplicative), and we used this function to give an expression for the product of divisors of a number.
Lecture 14: Divisors: A Summing Up
We started class by going over the midterm grades. We then discussed $\sigma$, the function which adds up the divisors of a given integer n. We saw that $\sigma$ was multiplicative and we gave a formula for evaluating $\sigma(n)$ based on a prime factorization of n. We then collected together a few interesting results about this important function.
Lecture 15: Perfect Numbers
In today's class, we discussed perfect numbers: those numbers n so that $\sigma(n) = 2n$. We gave a characterization of even perfect numbers and talked about properties that odd perfect numbers have — even though we don't know whether odd perfect numbers actually exist. We then briefly mentioned amicable pairs.
Lecture 16: Mu and Convolution
Today we began class by discussing some of the relevant details for the upcoming Group Projects. We then talked a little more about odd perfect numbers. The majority of the class was focused on defining convolution of arithmetic functions, viewing some of the results we already know in the light of these convolutions, and then using an important convolution identity to state and prove the Mőbius Inversion Formula (MIF).
Lecture 17: MIF; Quadratic Residues
In today's class we finished talking about convolutions, with the highlight being the proof and a few applications of the Mobius Inversion Formula. Afterwards we talked about quadratic congruences, ultimately defining and playing around with so-called quadratic residues.
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