Mathematics 5 Analytic Number Theory Spring 2025
Assignment 4
Please handin by 12 noon on Wednesday, 02 April
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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
1 p≡amod q
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
p≡amod q
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.
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1. Show that if θq (x) ∼ x, then (1) holds by completing the following steps. Fix any 0 < ϵ < 1.
a. Show that
ϕ(q)(1 − ϵ)[π(x;q) − π(x1−ϵ;q)]log x ≤ θq (x) ≤ ϕ(q)[π(x;q)]log x.
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 ,
2
holds for large x. Hence show that π(x;q) ~ x/[ϕ(q)log x].
2. Show that if the limit
3. Let F(z) := f(t)e−ztdt where f(t) = θq (et )e−t − 1. Show that for Re z > 1,
Hence show that
when Rez > 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 hypothe- ses of the theorem.