Hello, if you have any need, please feel free to consult us, this is my wechat: wx91due

Mathematics 5

Analytic Number Theory

Spring 2025

Assignment 4

Please handin by 12 noon on Wednesday, 02 April

The Prime Number Theorem for Primes in Arithmetic Progression

For two integers a and q with q ≥ 2 and gcd(a, q) = 1, consider the arithmetic progression APa,q = {a + mq : m ∈ Z}. The goal of this assignment is to show that

counts the number of primes p ∈ APa,q which are no greater than x. Here ϕ is the Euler totient function.

We adapt D. Newman’s proof of the PNT which is given in lecture. To do this, we introduce the functions

where 1 is the indicator function for APa,q. An important step in the proof is the analytic continuation of

to some open set containing {z : Re z ≥ 1}. This was worked out in Workshop 5 and you may use this without proof.

1. Show that if θq(x) ∼ x, then (1) holds by completing the following steps. Fix any 0 < ϵ < 1.

a. Show that

b. Show that (1 − ϵ)π(x1−ϵ; q) log x ≤ (1 − ϵ)x1−ϵ log x and hence deduce

c. Show that if θq(x) ∼ x, then for any b1 < 1 < b2,

holds for large x. Hence show that π(x; q) ∼ x/[ϕ(q) log x].

2. Show that if the limit

3. Let  Show that for Re z > 1,

Hence show that

when Re z > 0.

4. Show that F in Question 3 can be analytically continued to some open set containing {Re z ≥ 0} and complete the proof that (1) holds.

Hint: Use a theorem from the Lecture Notes but make sure you verify the hypotheses of the theorem.