Mathematics 5 Analytic Number Theory 2022/23 Assignment 3
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Mathematics 5
Analytic Number Theory
Assignment 3
2022/23
Please hand in by 12 noon on Friday, 18 November.
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For two integers a and q with gcd(a, q) = 1, consider the arithmetic progression APa,q = {a + mq : m ∈ 勿}. The goal of this assignment is to show that
counts the number of primes p ∈ Aa,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. Also you may appeal to Workshop problems without reproducing the proofs. To accomplish this, we introduce the following functions: set
In this assignment, any sum p←B or product p←B denotes a sum or product over all primes p ∈ B . If no B is explicit, then the sum or product is taken over all primes. Also any sum χ denotes a sum over all Dirichlet characters mod q . |
1. Let b denote any integer such that ba = 1 mod q . Show that
χ(a)χ(p) = χ(bp) = φ(q) ✶(p)
χ χ
where ✶ is the indicator function of APa,q and hence deduce that
✶(p) log p
pz .
Hint: See Workshop Sheet 3.
2. Show that
Φq,a (z) _ 
2
can be analytically continued to some open set Y Split the sum defining Φq,a (z) = χ〇 (a)ψ(z; χ〇 ) + Workshop Sheet 4.
containing {Re z > 1}. Hint:
χ
χ6 χ(a)ψ(z; χ) and review
3. Show that if θq (x) ~ x, then (1) holds by completing the following steps. Fix any 0 < e < 1.
a. Show that φ(q)(1 _ e)[π(x; q) _ π(x1 –e ; q)] log x < θq (x) < φ(q)[π(x; q)] log x.
b. Show that (1 _ e)π(x1 –e ; q) log x < (1 _ e)x1 –e log x and hence deduce θq (x) < φ(q)π(x; q) log x < 
θq (x) + 
φ(q)x log x.
c. Show that if θq (x) ~ x, then for any b1 < 1 < b2 ,
φ(q) log x
x
holds for large x. Hence show that π(x; q) ~ x/[φ(q) log x].
4. Show that if the limit
R θq (x) _ x
R玄女 1 x2
5. Let F (z) := 〇女 f(t)e –zt dt where f(t) = θq (e )et –t _ 1 and suppose that F has an analytic continuation to some open set Y which contains {z : Re z > 0}. Show
that the limit
R θq (x) _ x
R玄女 1 x2
Hint: You may use a theorem from lecture as long as you verify the hypotheses of the theorem.
6. Show that for Re z > 1,
φ(q) ✶(p) log p = z 女 θq (u) du = z 女 f(t)e –t4z – 1| dt + z
p
where f(t) = θq (et )e –t _ 1. Hence show that
F (z) = 
_ 
= ← Φq,a (z + 1) _ 
] 
_ 
when Re z > 0. Here F is defined in Exercise 5. Hint: see Exercise 1.
7. Let F be as in Exercise 5. Show that F can be analytically continued to some open set Y containing {Re z > 0}.
8. Put everything together and deduce that (1) holds.
2022-11-14