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 12 - An Intro to Arithmetic Functions
In today's class we finished our discussion of Euler's $\phi$ function, particularly how one goes about computing $\phi(n)$ for an arbitrary integer n. This left us considering a more general class of functions which share some of the properties of the $\phi$ function. We started investigating these so-called arithmetic functions, and finished with a proposition which gave us one method for constructing a "nice" arithmetic function out of an old "nice" arithmetic function.
Lecture 13 - Counting Divisors
In today's class we continued our discussion of arithmetic functions by looking at the function $\nu$ (pronounced "new"). This function counts the number of divisors of a given integer n, and it shares some of the abstract properties that our old friend $\phi$ enjoys.
Lecture 14 - Adding Up Divisors
We started today's class with a little observation of how number theory creeps into our daily lives without our realizing. We then spent the lion's share of the class discussing the function $\sigma$ which adds up the divisors of a given integers. We gave a formula which lets us compute $\sigma(n)$ provided we have a prime factorization of n. We also saw a handful of results which described some properties of this 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
We started today's class by introducing a few new arithmetic functions. When then spent the balance of the class period 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 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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