MATH 568 NUMBER THEORY II, SPRING 2012, PROBLEMS 2
Due Tuesday 24th January
1. (i) Prove that
x
1
(u)
du = log x + O(1).
u2
(ii) Prove that lim supx (x) 1 and lim inf x (x) 1.
x
x
(iii) Prove that if there is a constant c such that (x) cx as x , then c =
Math 568 Number Theory II, Spring 2012, Problems 3
Due Tuesday 31st January
n/d
n
1. (i) Let f A. Prove that
f (r ) =
(d)
s=1
d|n
r =1
(r,n)=1
f (ds).
We use e() to denote exp(2i). (ii) Prove that
n
e
r =1
n
when n|m,
0
m
r=
n
when n n.
n
Let cn (m) denot
Math 568 Number Theory II, Spring 2012, Problems 4
Due Tuesday 7th February
1. (cf Hille (1937) Suppose that f (x) and F (x) are complex-valued functions dened on [1, ).
Then
F (x) =
f (x/n)
nx
for all x if and only if
f (x) =
(n)F (x/n)
nx
for all x.
2.
MATH 568, NUMBER THEORY II, SPRING 2012, PROBLEMS 7
Due Tuesday 28th February
The whole of this homework is due to Ingham (1929). Homework 5.2 will be useful.
1. Suppose that is a non-zero real number and that (1 + i ) = 0. Let
|i (n)|2 ns
f (s) =
n=1
and
MATH 568 NUMBER THEORY II, SPRING 2012, PROBLEMS 1
Due Tuesday 17th January
1. Given a|b and c|d, prove that ac|bd.
2. Prove that if n is odd, then n2 1 is divisible by 8.
3. Find the greatest common divisor g of the numbers 1819 and 3587, and then
nd int
Math 568, Number Theory II, Spring 2012, Problems 13
Due Tuesday 17th April
1. Prove that for any xed A > 0,
p x
1
where c =
p
(p 1)
= cli(x) + O x(log x)A
p1
1
.
p(p 1)
2. (i) Prove that if n N, then
n
=
(n)
d|n
(d)2
.
(d)
(ii) Prove that for any xed A >
MATH 568, NUMBER THEORY II, SPRING 2012, PROBLEMS 11
Due Tuesday 3rd April
1. Prove that if > 0, then
(s)
Deduce that if (s) = 0, then
s
|s|
<
.
s1
1+t2
2.
2. It is convenient to employ the notation
E0 () =
1
if = 0 ,
0
otherwise.
Prove that if is a ch
Math 568, Number Theory II, Spring 2012, Problems 9
Due Tuesday 20th March
Throughout is a non-principal character modulo p and we use the notation and results
of homework 8.2 (with q = p). It is also useful to observe that (1) = (1) = 1.
p 1
2
|S (p, a,
MATH 568, NUMBER THEORY II, SPRING 2012, PROBLEMS 10
Due Tuesday 27th March
Throughout denotes a non-principal character modulo q , M denotes a nonnegative
integer,
S (x; ) =
(n),
(n)ns
L(s; ) =
M <nx
n=1
and s = + it is a complex number with > 0. For bre
568 NUMBER THEORY II, SPRING TERM 2012, PROBLEMS 14
Return by Tuesday 24th April
1. Show that if 1 , . . . , n are real numbers and R 2 is an integer, then there
are a1 , . . . , an and q with 1 q Rn 2n + 1 such that
|1 a1 /q | q 1 R1 , . . . , |n an /q |
Math 568, Number Theory II, Spring 2012, Problems 11
Due Tuesday 10th April
This homework is a continuation of the previous one. Here we investigate the zerofree region
for Lfunctions formed from quadratic characters. We suppose throughout that is real bu
MATH 568, NUMBER THEORY II, SPRING 2012, PROBLEMS 6
Due Tuesday 21st February
x
1
w
1. Let b1 (x) = x x 1 , bn+1 (x) = 0 bn (y )dy 0 0 bn (y )dy dw. (i) Show that
2
1
b (y )dy = 0 and that bn has period 1.
0n
(ii) Show further that if n > 1, or n = 1 and
Math 568 Number Theory II, Spring 2012, Problems 5
Due Tuesday 14th February
(n)ns ,
1. Determine
2. Let a (n) =
d|n
(n)ns , and
|(n)|ns in terms of the zeta function.
da . Show that
a (n)b (n)ns = (s) (s a) (s b) (s a b)/ (2s a b)
n=1
when > max 1, 1 +