PMATH 740 Analytic Number Theory, Assignment 3
log n
=
n
nx
(n)
(b) Show that
=
n
1: (a) Show that
1
2
log2 x + c + O
1
2
Due Fri June 22
log2 x + 2 log x + O(1).
log x
x
for some constant c.
nx
(c) Show that for 1 < a R we have
nx
log x
(n)
=
+ (a)2 +

The Value of the Zeta Function at Positive Even Integers
Theorem: For n Z+ we have
1
(2n) = 2 2n c2n
where c2n is the coecient of x2n in the Talylor series at 0 for z cot z .
cot z
Proof: Let f (z ) = 2n . Note that f is holomorphic in C except at the po

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PMATH 740, Analytic Number Theory
Midterm Test, Spring Term, 2012
University of Waterloo
Instructor: Stephen New
Date:
June 27, 2012
Time:
4:00-6:00 pm
Instructions:
Question
1. Place your name, signature and ID number

PMATH 740 Analytic Number Theory, Solutions to Assignment 3
1: (a) Show that
nx
log n
=
n
1
2
log2 x + c + O
log x
x
for some constant c.
Solution: We use Eulers Summation Formula with f (t) =
nx
x
log n
= f (1) +
n
1
log t
dt +
t
log2 t
+
1
1
1
2
=
where

PMATH 740 Analytic Number Theory, Solutions to Assignment 2
1: (a) Find the number of cubes, and the number of twelfth powers in U81 .
Solution: Recall that for an element a with ord(a) = n in a nite group G, we have ak = ad where
d = gcd(k, n) and ord(ak

PMATH 740 Analytic Number Theory, Solutions to Assignment 1
1: Let a = (25)! and b = (5500)3 (1001)2 .
(a) Find the prime factorization of a and of b.
Solution: Recall that the exponent of the prime p in n! is ep (n!) =
n
p
+
a
p2
+
a
p3
, so we have
a =

PMATH 740 Analytic Number Theory, Course Outline
Lectures: MWF 1:30-2:20 in MC 4064.
Instructor: Stephen New, oce MC 5163, extension 35554, oce hours MW 2:40-4:00.
Text: Introduction to Analytic Number Theory, by Apostol.
Course Outline: We will try to co

1. Groups
1.1 Denition: A ring is a set R with elements 0, 1 R and operations +, : R2 R
such that
(1) (a + b) + c = a + (b + c) for all a, b.c R,
(2) a + b = b + a for all a, b R,
(3) a + 0 = a for all a R,
(4) for all a R there exists a unique b R with a

PMATH 740 Analytic Number Theory, Assignment 5
Not to hand in
1: Let l Z+ and let 1 denote the identity character in Ul .
1
1 pz .
(a) Show that L1 (z ) = (z )
1
p|l
(b) Note that L1 (z ) has a simple pole at 1. Find the residue at this pole.
1
1 n)(n)
1(

PMATH 740 Analytic Number Theory, Assignment 4
1: (a) Show that
p x
Due Fri July 20
1
converges.
p log p
(b) Determine whether
n=pk x
1
converges.
n (n)
2: (a) Let : Z C be periodic, completely multiplicative and not identically zero. Let l
be the smalles

PMATH 740 Analytic Number Theory, Assignment 2
Due Fri June 8
1: (a) Find the number of cubes, and the number of twelfth powers in U81 .
(b) Find the number of cubes, and the number of twelfth powers in U128 .
(c) For n = 18900, nd the universal exponent

PMATH 740 Analytic Number Theory, Assignment 1
Due Fri May 25
1: Let a = (25)! and b = (5500)3 (1001)2 .
(a) Find the prime factorization of a and of b.
(b) Find the prime factorization of gcd(a, b) and of lcm(a, b).
(c) Find the number of positive factor