Question 18.0.1.
What properties do number-theoretic functions (such as \(\phi(n)\)) have? What can we do with them?
Chapter 18 An Introduction to Functions
The further one goes into number theory, the more one needs to think about the functions involved as functions, and not just as handy computational shorthand.
Most of the remainder of the text deals with such questions. This short chapter introduces some of the questions we will ask through the lens of one function we have done a fair amount with, and then through the eyes of one we have examined in less detail.
The Euler function, like many we have seen and will see, is an example of an arithmetic function. An arithmetic function is a function with the positive integers as its domain, usually going to integer, real, or complex values.
Remark 18.0.2.
We pronounce this word with the stress on the third syllable in number theory when used as an adjective, but (as usual) on the second syllable when used as a noun.
A-rith-me-tic functions show up when studying the higher a-rith-me-tic.
We’ll spend a lot of time with three types of questions regarding arithmetic functions. For any given function, we wish to find or examine the following.
- We want to have as explicit formulas as possible for our functions, which are often defined implicitly or in terms of counting.
- We wish to find relational formulas, either between our function and other functions, or especially among different values of the function itself.
- We desire to see what the long-term or aggregate behavior of the functions is; in practice this usually involves summation of various kinds.
In this chapter, we will start the process, but it will recur throughout the remainder of the text.
Summary: An Introduction to Functions
This short chapter introduces us to arithmetic functions, and raises some interesting questions we can ask about them.
- In Section 18.1 we review the formula and relations for the familiar Euler \(\phi\) function, while also asking it where it is going.
- In Section 18.2 we ask the same questions of a new function, which culminates in the surprising Fact 18.2.9.
The Exercises just give a little chance to think about functions, in preparation for the next several chapters.
