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Number Theory: In Context and Interactive

Karl-Dieter Crisman

Chapter 23 New Functions from Old

We are heading toward the end of the text. There are even more interesting functions out there; just as important, there are more interesting ways to start connecting these functions to calculus.
As a prelude, let us introduce an interesting function. Letting \(p\) be running just over primes, we let
\begin{equation*} D(N)=\prod_{p\leq N}\left(1-\frac{1}{p}\right) \end{equation*}
and then expand the expression as a sum of unit fractions. As an example,
\begin{equation*} D(3)=(1-1/2)(1-1/3) = \left(\frac{1}{1}-\frac{1}{2}-\frac{1}{3}+\frac{1}{6}\right)\text{.} \end{equation*}
Before starting this chapter, try expanding \(D\) (as above, without adding the fractions) for bigger and bigger values of \(N\text{.}\) What patterns do you find?
  • What denominators show up?
  • Which ones don’t?
  • For the ones that do, what are the values of the numerator?
  • Can you predict the value of the numerator for some types of denominators? (E.g., primes, perfect squares, prime powers, etc.)
The function unveiled by this is quite important in expanding our roster of arithmetic functions and unlocking their secrets, as well as in connecting to calculus.
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