A PRIMER OF ANALYTIC NUMBER THEORY: From Pythagoras to ...

Chapter 2
Products and Divisibility
Iam one who becomes two
Iam two who becomes four
Iam four who becomes eight
Iam the one after that
Egyptian hieroglyphic inscription from the 22nd dynasty (Hopper, 2000)
2.1. Conjectures
Questions about the divisors, d, of an integer n are among the oldest in
mathematics. The divisor function �(n) counts how many divisors n has.
For example, the divisors of 8 are 1, 2, 4, and 8, so �(8) = 4. The divisors
of 12 are 1, 2, 3, 4, 6, and 12, so �(12) = 6. The sigma function �(n) is
defined as the sum of the divisors of n. So,
�(8) = 1 + 2 + 4 + 8 = 15,
�(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28.
In the Sigma notation of Chapter 1,
�(n) = �
d.
The difference here is that we are summing not over a set of consecutive
integers but only those d which divide n, as the subscript d | n indicates.
Similarly,
�(n) = �
1.
Here, we add to our count a 1, not d, for each divisor d of n.
Exercise 2.1.1. Isaac Newton computed how many divisors 60 has in his
1732 work Arithmetica Universalis. What is �(60)?
24
d|n
d|n