Exercise 4.1.4

You should first establish the following inequality for n = (p1)^(k1) ... (pr)^(kr):

sig(n)/n < (p1/(p1-1)) ... (pr/(pr - 1)). 

To do this, begin by writing out the formula for sig(n) as a product of factors involving ( (pi)^(ki + 1) - 1) / (pi - 1).  Dividing this product by n in prime factored form allows you to consider factors of the form ( (pi)^(ki + 1) - 1) / (pi)^(ki).  Each of these factors can't be any bigger than pi.  Why?

Once you've established the inequality discussed, in a new paragraph, turn your attention to the factors p / (p - 1) and q / (q - 1) where p and q are the primes that we're assuming describe the odd perfect n.   Since n is perfect, 2 must equal sig(n)/n, which is less than (p / (p - 1))(q / (q - 1)) by the fact above.  On the other hand, just how big can these last two factors be?  Just how big can their product be?  Be sure to prove your estimates.  A guess based on consideration of several sample computations does not constitute a proof.