Chapter 1
Introduction and classic methods
Exercise 1.1. If (a, b) = 1 and if c | a and d | b, then (c, d) = 1.
Exercise 1.2. If (a, b) = (a, c) = 1, then (a, bc) = 1.
Exercise 1.3. If (a, b) = 1, then (an , bk ) = 1 for all n ≥ 1, k ≥ 1.
Exercise 1.4. If (a, b) = 1, then (a + b, a − b) is either 1 or 2.
Exercise 1.5. If (a, b) = 1, then (a + b, a2 − ab + b2 ) is either 1 or 3.
Exercise 1.6. If (a, b) = 1 and if d | (a + b), then (a, d) = (b, d) = 1.
Exercise 1.7. A rational number a/b with (a, b) = 1 is called a reduced fraction. If the sum
of two reduced fractions is an integer, say (a/b) + (c/d) = n, prove that |b| = |d|.
Exercise 1.8. An integer is called squarefree if it is not divisible by the square of any prime.
Prove that for every n ≥ 1 there exist uniquely determined a > 0 and b > 0 such that n = a2 b,
where b is squarefree.
Exercise 1.9. For each of the following statements, either give a proof or exhibit a counter
example.
1. If b2 | n and a2 | n and a2 ≤ b2 , then a | b.
2. If b2 is the largest square divisor of n, then a2 | n implies a | b.
Exercise 1.10. Given x and y, let m = ax + by, n = cx + dy, where ad − bc = ±1. Prove that
(m, n) = (x, y).
Exercise 1.11. Prove that n4 + 4 is composite if n > 1.
Exercise 1.12. Prove that if 2n − 1 is prime, then n is prime.
Exercise 1.13. Prove that if 2n + 1 is prime, then n = 0 or n is a power of 2.
Exercise 1.14. Find all integers n such that
(a) ϕ(n) = n/2,
(b) ϕ(n) = ϕ(2n)
(c) ϕ(n) = 12.
Exercise 1.15. For each of the following statements either give a proof or exhibit a counter
example.
1. If (m, n) = 1 then (ϕ(m), ϕ(n)) = 1.
1
2
CHAPTER 1. INTRODUCTION AND CLASSIC METHODS
2. If n is composite, then (n, ϕ(n)) > 1.
3. If the same primes divide m and n, then nφ(m) = mφ(n).
Exercise 1.16. Let ζ(s) be the Riemann zeta function defined as the associate Dirichlet series
of the arithmetic function 1(n) = 1 (n ≥ 1). Express with help of the zeta-function the
functions
1. µ(n)
2. µ(n)2
3. ϕ(n)
4. σ(n)
5. τ (n)
6. 2ω(n)
7. Λ(n)
Exercise 1.17. Show that for all n ≥ 1, we have that 2ω(n) ≤ τ (n) ≤ 2Ω(n) .
Exercise 1.18. Let f be an arithmetic function with real values. We suppose that there exist
two prime numbers p and q such that f (pq)2 < 4f (p2 )f (q 2 ). Show that f is either invertible
or a prime element of Ar the ring of real valued arithmetic functions.
Exercise 1.19. Let Λ be von Mangoldt’s function. Then prove that
P
(a)
d|n Λ(d) = ln n
x
P
(b)
d≤x Λ(d) d = x ln x − x + O(ln x).
Exercise 1.20. Let pn denote the n-th prime and dn = pn+1 − pn . Suppose that the prime
number theorem holds in its elementary form
π(x) ∼
x
ln x
(x → ∞).
Prove the following assertions:
(a) pn ∼ n ln n (n → ∞).
P
(b)
1<n≤x dn / ln n ∼ x (x → ∞).
(c) lim inf n→∞ (dn / ln n) ≤ 1 ≤ lim supn→∞ (dn / ln n).
Exercise 1.21. Recall that Λ is the von Mangoldt function and ψ(x) is its summatory fuction.
P
(a) Show that n≤x (ln n)/n = 12 (ln x)2 + O(1) (x ≥ 1).
(b) Show that
X Λ(n) Z x ψ(t)
ψ(x)
=
dt +
2
n
t
x
1
n≤x
(x ≥ 1).
3
(c) We suppose that the prime number theorem in its strong form
x
ψ(x) = x + O
(ln 2x)2
holds. Show the existence of a constant A such that
X Λ(n)
= ln x + A + o(1)
n
(x → ∞).
n≤x
(d) Show that the prime number theorem in its strong form implies the following asymptotic
relation
X ln p
= ln x − γ + o(1).
p−1
p≤x
Furthermore also prove the contrary, i.e. this relation implies the prime number theorem.
Exercise 1.22. Let FN be the Farey-sequences of order N and a/b, a00 /b00 and a0 /b0 three
consecutive elements of FN appearing in this order. Show that
a00
a + a0
=
.
b00
b + b0
P
Exercise 1.23. Let A(n) := pν ||n νp be the Alladi-Erdős function. Suppose that the prime
number theorem holds in the form
x x
π(x) =
.
+ O (R(x)) , R(x) = o
ln x
ln x
Show that
X
n≤x
A(n) ∼
π 2 x2
12 ln x
(x → +∞).
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CHAPTER 1. INTRODUCTION AND CLASSIC METHODS
Chapter 2
Methods from complex analysis
Exercise 2.1. The Theorem of Bohr and Mollerup Let ϕ : R → R be a convex function
such that ϕ(1) = 0, ϕ(x + 1) = ϕ(x) + ln x (x > 0).
(a) Show that for every integer n ≥ 1, the function
ψn (x) := {ϕ(n + 1 + x) − ϕ(n + 1)}/x
is monotonically increasing on [−1, 0[∪]0, 1].
(b) Deduce from (a) that


Y
0 ≤ ϕ(x) − ln n!nx /
(j + x) ≤ 1/n
(0 < x ≤ 1, n ≥ 1).
0≤j≤n
(c) Show that ϕ is uniquely determined and satisfies Gauss’ Formula
Y
eϕ(x) = lim n!nx
(j + x)−1 (x > 0).
n→∞
0≤j≤n
Exercise 2.2. Show that
Z 1
Γ(x)Γ(y)
(1 + t)x−1 (1 − t)y−1 dt = 2x+y−1
Γ(x + y)
−1
P
Exercise 2.3. Let A(t) := n≤t (−1)n n (t ≥ 0).
(x > 0, y > 0).
(a) Show that A(N ) = (−1)N b 21 (N + 1)c for every integer N ≥ 1.
R +∞
(b) Calculate the abscissa of convergence of G(s) := 1− t−s−1 dA(t).
(c) Show that for every σ > 0 we have
Z
G(s) = (s + 1)
1
(d) Show that the series
R n+1
dt/t1+ε
n≥1 A(n) n
P
5
∞
A(t)
dt.
ts+2
converges for every ε > 0.
(2.1)
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CHAPTER 2. METHODS FROM COMPLEX ANALYSIS
(e) Deduce that (2.1) defines a holomorphic continuation of G(s) to the half-plane σ > −1.
P
Exercise 2.4. Let F (s) = n≥1 an n−s be a Dirichlet-series with abscissa of convergence σc .
P
(a) Show that the series ϕ(z) = n≥1 an e−nz converges uniformly for <z ≥ ε > 0.
(b) Let H := {s : σ > max(0, σc )}. Show that for every s ∈ H, there exists a constant C(σ)
−σ
such that |ϕ(t)| ≤ C(σ)t
R ∞ s−1 (t > 0). Deduce that the domain of absolute convergence of
the integral I(s) = 0 t ϕ(t)dt contains the half-plane H.
(c) Show that for s ∈ H, we have that F (s)Γ(s) = I(s).
Exercise 2.5. Use the Euler-Maclaurin-Formula with f (t) := t−s to show that for s > 1 we
have
X Br+1 s + r − 1 s + k Z ∞
s
+
ζ(s) =
−
Bk+1 (t)t−s−k−1 dt.
s−1
r+1
r
k+1 1
0≤r≤k
Deduce another proof of the analytic continuation of ζ and show the trivial zeros of ζ.
P
Exercise 2.6. Let (an )n≥0 ∈ RN and b ∈ R. For n ≥ 0 we set sn := 0≤m≤n am .
P
(a) Calculate bN := (1/N ) 0≤n≤N sn as function of a0 , . . . , aN . Deduce that under the
hypotheses
lim bN = b,
(2.2)
N →∞
aPnecessary and sufficient condition for the convergence of the series
0≤n≤N nan = o(N ) for N → ∞.
P
n≥0 an
to b is that
(b) We suppose that (2.2) holds.
P
(i) Calculate sn P
as a function of bn and bn−1 . Show that the series S(z) := n≥0 sn z n
and A(z) := n≥0 an z n converges in the open unit disk.
P
(ii) Show that S(x) = (1 − x) n≥0 nbn xn for 0 < x < 1. Deduce that limx→1− (1 −
x)S(x) = b. One could
set bn = b + εn (n ≥ 0), where εn tends to 0 and bound
P
n
E(x) := n≥0 εn nx .
(iii) For 0 < x < 1 write (1 − x)S(x) as a function of A(x).
Exercise 2.7. (a) Use directly the integral definition of Γ to show that
Z
1 ∞
0
−Γ (1) = ln x −
(ln t)e−t/x dt (x > 1).
x 0
(b) Use the approximation ln t =
P
n≤t 1/n
− γ + O(1/t) for t > 1 to show that
Γ0 (1) = −γ.
Exercise 2.8. Let F (s) =
with <z > −1 such that
P
n≥1 an n
A(x) :=
−s
be a Dirichlet series. Suppose that there exists a z ∈ C
X
1≤n≤x
an ∼ x(ln x)z
(x → ∞).
7
Show that
F (s) =
Γ(z + 1)
+o
(s − 1)z+1
1
(σ − 1)<z+1
if s → 1 in the half-plane σ > 1.
P
Exercise 2.9.
1. The series n n−s does not converge in any point of the line {s = 1+it, t ∈
R}.
P
2. The series p p−s does converge in every point s = 1 + it with t 6= 0. Use the prime
number theorem of the form
x
x
π(x) =
.
+O
ln x
(ln x)2
Exercise 2.10. For κ > max(0, σc ) show that
X
an
n≤x
x k
k!
ln
=
n
2πi
Z
κ+i∞
F (s)xs
κ−i∞
ds
,
sk+1
where k is an arbitrary positive integer.
Exercise 2.11. Let τk (n) denote the number of solutions of n = m1 · · · mk in integers m1 , . . . , mk ≥
1.
(a) Show that
X
τk (n)n−s = ζ(s)k .
n≥1
(b) Show that for every k ≥ 1, there exists a δk > 0, which we can calculate explicitly, such
that
X
τk (n) = xPk−1 (ln x) + Oε x1−δk +ε ,
n≤x
where Pk−1 is a polynome of degree k − 1.
Exercise
2.12. Let {ap } be a bounded sequence indexedPby the prime numbers. Show that,
P
−σ
if
ap p tends to a limit for σ → 1+, then the series
ap p−1 converges.
Show that this is false for a sequence indexed by the positive integers.
Exercise 2.13. Calculate all Dirichlet characters mod 15.
Exercise 2.14. Show all homomorphisms of the group (Zk , +) into (C∗ , ·). Do there also exist
some orthogonality relations?
Exercise 2.15. Deduce from the prime number theorem in arithmetic progressions that there
are infinitely many primes whose decimal expansion starts with a 1 and ends with 7.