Chapter 23 New Functions from Old
D(N)=∏p≤N(1−1p)
and then expand the expression as a sum of unit fractions. As an example,
D(3)=(1−1/2)(1−1/3)=(11−12−13+16).
Before starting this chapter, try expanding D (as above, without adding the fractions) for bigger and bigger values of N. 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.)