189-346/377B: Number Theory Assignment 5

Due: Monday, March 21

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 (or otherwise), show that the number of squarefree 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 + 3 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. Let d be a prime. Show that any prime p which does not divide d but divides the integer

n^{d−1} + n^{d−2} + ⋯ + 1

(n ∈ Z) is necessarily of the form 4d + 1. Use this to show that there are infinitely many primes of the form 4d + 1. (Hint: assume otherwise, and study the asymptotics of #{n^{d−1} + ⋯ + n + 1, n ≤ x^{1/d}} as x⟶∞ in two different ways to derive a contradiction.)

The following exercises are taken from the textbook by Levesque.

5. (Section 6.2, exercise 7 from Levesque.) Show that, for all s > 1,

∑_{n=1}^{∞} \frac{1}{n^{s}} · ∑_{n=1}^{∞} \frac{μ(n)}{n^{s}} = 1,

where μ(n) is the Möbius function defined by μ(n) = (−1)^{t} if n is a product of t distinct primes, and μ(n) = 0 if t is divisible by the square of some prime.

6. Show that if f(x) is a continuous, monotonically decreasing function which tends to 0 as x⟶∞, and if the series ∑_{n=1}^{∞} f(n) diverges, then the function

F (n) := ∑_{j=1}^{n} f(j)

satisfies

F (n) ∼ ∫_{1}^{n} f(x)dx.

7. (Section 6.4, exercise 9 from Levesque.) Let log_{k} x be the k-th iterate of the logarithm function, defined recursively by

log_{1} x = log x, log_{k} x = log log_{k−1} x.

Is there a continuous increasing function f(x) such that lim_{x→∞} f(x) = ∞, yet f(x) = o(log_{k} x) for all k ≥ 1? If so, exhibit such a function.

Math 377 only:

8. Section 6.8., exercise 4 in Levesque.