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The University of Bristol's examination paper for the Analytic Number Theory module (MATH-M0007) at Level M, held in Summer 2019. The paper covers topics such as Dirichlet convolution, multiplicative arithmetic functions, Gauss sums, and the Riemann zeta function. Students are required to answer four questions within 2 hours and 30 minutes, without the use of calculators.
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Examination for the Degrees of B.Sc. and M.Sci. (Level M)
ANALYTIC NUMBER THEORY MATH M (Paper Code MATH-M0007)
Summer 2019 2 hours and 30 minutes
This paper contains four questions All four answers will be used for assessment. Calculators are not permitted in this examination.
On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.
Do not turn over until instructed.
I(n) =
1 if n = 1, 0 if n > 1, 1 (n) = 1,
μ(n) =
1 if n = 1, (−1)r^ if n is a product of r distinct primes, 0 otherwise, ϕ(n) = #{a ∈ N : a ≤ n and gcd(a, n) = 1}.
(a) i. (2 marks) Define the Dirichlet convolution f ∗ g of two arithmetic functions f and g. ii. (2 marks) Let L(n) = log n. Show that
L · (f ∗ g) = (L · f ) ∗ g + f ∗ (L · g).
Here · is the usual multiplication. iii. (2 marks) Define what it means for an arithmetic function to be multiplicative. iv. (2 marks) Show that the Dirichlet convolution of two multiplicative arithmetic functions is multiplicative. v. (2 marks) Prove that μ ∗ 1 = I. vi. (3 marks) Ignoring issues of convergence, show that ∑^ ∞ n=
μ(n) ns^ =^
ζ(s).
(b) Define N (n) = n. i. (2 marks) Show that ϕ = μ ∗ N. ii. (3 marks) Deduce that ϕ is multiplicative and prove that
ϕ(n) = n
p|n
1 − (^1) p
iii. (2 marks) Prove that ϕ ∗ 1 = N. iv. (5 marks) Prove that ∑
n≤x
ϕ(n) n =^
π^2 x^ +^ O(log^ x).
(Use ζ(2) = π 62 .)
(c) Let χ be a primitive Dirichlet character modulo q and let Sχ(N ) =
n≤N
χ(n)
for any N > 0. Let L(s, χ) =
n=
χ(n) ns^. Recall the P´olya-Vinogradov inequality, which states that there is an absolute constant c > 0 such that |Sχ(N )| ≤ c√q log q. i. (3 marks) Use partial summation to show that
L(s, χ) = s
1
y−^1 −sSχ(y)dy.
ii. (5 marks) Let s = σ + it for σ ∈ (1/ 2 , 1). Deduce that L(s, χ) |s|q 1 −^2 σlog q.
Γ(s) =
0
xs−^1 e−x^ dx,
for <(s) > 0. i. (2 marks)Using the identity ∫ (^) ∞
−∞
2 π
e−y^2 /^2 dy = 1
or otherwise, show that Γ(1/2) = √π. ii. (8 marks) Explain why Γ(s) is analytic in the region {s ∈ C : <(s) > 0 } and show how it can be extended to a meromorphic function on C. What is the relationship between Γ(s) and n!? iii. (3 marks) Show that Γ(s) has a simple pole at s = 0 with residue 1. (b) Let ζ(s) be the Riemann zeta function. i. (4 marks) Prove that for <(s) > 1, we have
ζ(s) = s s − 1
− s
1
{x} xs+^
dx,
where {x} denotes the fractional part of x. ii. (1 mark) State the Riemann Hypothesis for the Riemann zeta function. iii. (2 marks) Show why there are no zeros of the Riemann zeta function in the region {s ∈ C : 1 < <(s)}. iv. (5 marks) Using the functional equation
π−^ s^2 Γ
( (^) s 2
ζ(s) = π s−^2 1 Γ
1 − s 2
ζ(1 − s),
or otherwise, show that ζ(0) = −^12.
Λ(n) =
log p, if n is a power of prime p, 0 , otherwise.
Show that (^) ∑
n≤x
Λ(n) n
p≤x
log p p
ii. (4 marks) Using ∑ n≤x
Λ(n) n = log^ x^ +^ O(1) or otherwise, show that ∑
p≤x
p = log log^ x^ +^ O(1).
(b) (3 marks) Recall that
π(x) =
p≤x
1 , θ(x) =
p≤x
log p, ψ(x) =
n≤x
Λ(n).
State the Prime Number Theorem for these functions. (c) i. (5 marks) Let θ(y) =
n=−∞
e−πn^2 y^ y > 0.
Show that θ(y) = y−^12 θ(y−^1 ), clearly stating what conditions you have used. ii. (10 marks) Using this result, show that
π−^
s 2 Γ
( (^) s 2
ζ(s) =
n=
n−sπ−^
s 2 ∫^ ∞ 0
e−tt
s 2 − 1 dt
1
θ(1/y) − 1 2 y
− s 2 − (^1) dy +
1
θ(y) − 1 2 y^
s 2 − 1 dy.
From this, show that the functional equation is invariant under the transformation s → 1 − s, i.e. that
π−^
s 2 Γ
( (^) s 2
ζ(s) = π−^
1 − 2 s Γ
1 − s 2
ζ(1 − s).
End of examination.
