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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.

Question 18.0.1.

What properties do number-theoretic functions (such as \(\phi(n)\)) have? What can we do with them?
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.