507 – Homework 2

This set is due Monday, October 18.

1. Given arithmetic functions , define their convolution by
   

   so that if is the Dirichlet series associated to a function , then .

   Last homework we showed that , i.e., , where is the function that always takes the value , and is the function that takes value 1 at 1, and 0 otherwise.

   Prove the Möbius inversion formula: If , i.e., for all , then

   

   for all . (This is Theorem 6.14 in the text, try to prove it without looking at the argument there.)

   Use this and the fact shown in class that to find a formula for Euler’s .

   A function is multiplicative iff whenever are relatively prime. Let be the function defined by . Show that if is multiplicative, then so is .

   Also, solve exercise 8.2.5 from the book.

2. This exercise continues the analysis of from the first homework. First, by induction on , show that any that is not a unit and is not 0 is a product of irreducibles.Second, show that is a principal ideal domain (pid). This means that has no zero divisors and that every ideal is principal. You may want to look up the meaning of these terms in an abstract algebra book if needed. Here is a hint, it should remind you of the corresponding argument we proved for : Given an ideal , let be of minimal norm among the nonzero elements of . We want to show that consists precisely of the multiples of . For this, show that the with form the vertices of an infinite family of squares that fill up the plane. For example, one of these squares has vertices . Check that contains all these , and nothing else, or else a geometric argument shows that is not minimal.

   Use that is a pid to conclude that admits unique factorization.

   Here is a way of using these results to show Fermat’s two squares theorem: Let be a prime with . We know that the equation has a solution . Use that to conclude that is not irreducible in . By considering a product where neither nor is a unit, and taking norms, conclude that must be a sum of two squares.

3. The goal of this exercise is to show that the proof given in lecture of Fermat’s result discussed in the previous exercise generalizes to show:
   1. A prime is of the form iff or .
   2. A prime is of the form iff .

   First, check that for any ,

   

   We proved that if is sum of two relatively prime squares and so is the prime , and , then is itself sum of two relatively prime squares. State and prove the appropriate version of this result for each of the two cases we are now considering. Check that the argument works when and (although of course 4 is not prime).

   Show that the descent argument can be adapted to show that if a prime divides a number of the form with , then has the form .

   Show the similar result for . Be careful, since you must ensure that the prime that is obtained in this case is odd (because the result fails for ).

   Suppose that is prime. Prove that has a solution, by appropriately factoring .

   Conclude the proofs.

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