189-346/377B: Number Theory Assignment 5

Due: Monday, March 12

1. An integer n is said to be square-free if its prime factorisation is of the form

n = p_{1}p_{2} ⋯ p_{r},

where p_{1}, …, p_{r} are distinct primes. Show that for all real s > 1,

\frac{ζ(s)}{ζ(2s)} = ∑_{n∈S} \frac{1}{n^{s}} ,

where

ζ(s) = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + ⋯

is the Riemann zeta function, and S is the set of positive square free integers.

2. Using a Sieve argument, show that the number of square-free integers that are less than or equal to x is equal to

ζ(2)^{−1}x + o(x).

3. Show that any integer of the form 4n−1 always has a prime divisor of the form 4k + 3. Use this to give a proof that there are infinitely many primes of the form 4k + 3, analogous to Euclid’s proof of the infinitude of primes that was recalled in class. Show by a similar argument that there are infinitely many primes of the form 3k + 2.

4. Show that any prime p which divides the integer n^{2} + 1 (n ∈ Z) is either equal to 2, or is of the form 4k + 1. Use this to show that there are infinitely many primes of the form 4k + 1. (Hint: assume otherwise, and study the asymptotics of #\{n^{2} + 1, n ≤ \sqrt{x}\} as x⟶∞ in two different ways to derive a contradiction.) 377: Use a similar approach to prove that there are infinitely many primes of the form 3k + 1. Generalise your method to show that, if q is a fixed prime, there are infinitely many primes which are congruent to 1 modulo q. The following exercises are taken from the textbook by Leveque.

5. Section 6.2, exercise 7.

6. Section 6.4, exercise 5.

7. Section 6.4, exercise 6.

8. Section 6.4, exercise 9. 377:

9. Section 6.8., exercise 4.